stan::math Namespace Reference

Detailed Description

Matrices and templated mathematical functions.

Templated probability distributions. All paramaterizations are based on Bayesian Data Analysis. Function gradients via reverse-mode automatic differentiation.

Namespaces
	internal
	Returns the elementwise abs() of the input, which may be a scalar or any Stan container of numeric scalars.
	opencl
	opencl_kernels

Classes
struct	abs_fun
	Return elementwise absolute value of the specified real-valued container. More...
class	accumulator
	Class to accumulate values and eventually return their sum. More...
class	accumulator< T, require_var_t< T > >
	Class to accumulate values and eventually return their sum. More...
class	acos_
struct	acos_fun
	Structure to wrap acos() so it can be vectorized. More...
class	acosh_
struct	acosh_fun
	Structure to wrap acosh() so it can be vectorized. More...
struct	ad_promotable
	If the type From can be converted to To using implicit conversions, or both From and To are possibly cv-qualified void), provides the member constant value equal to true. More...
class	ad_tape_observer
	TBB observer object which is a callback hook called whenever the TBB scheduler adds a new thread to the TBB managed threadpool. More...
class	addition_
class	addition_operator_
class	adjoint_results_cl
	Represents results that are adjoints of vars in kernel generrator expressions. More...
class	append_col_
	Represents appending of cols in kernel generator expressions. More...
struct	append_return_type
	This template metaprogram is used to compute the return type for append_array. More...
struct	append_return_type< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > >
	This template metaprogram is used to compute the return type for append_array. More...
struct	append_return_type< int, int >
	This template metaprogram is used to compute the return type for append_array. More...
struct	append_return_type< std::vector< T1 >, std::vector< T2 > >
	This template metaprogram is used to compute the return type for append_array. More...
class	append_row_
	Represents appending of rows in kernel generator expressions. More...
struct	apply_scalar_unary
	Base template class for vectorization of unary scalar functions defined by a template class F to a scalar, standard library vector, or Eigen dense matrix expression template. More...
struct	apply_scalar_unary< F, fvar< T > >
	Template specialization to fvar for vectorizing a unary scalar function. More...
struct	apply_scalar_unary< F, std::vector< T > >
	Template specialization for vectorized functions applying to standard vector containers. More...
struct	apply_scalar_unary< F, T, require_complex_t< T > >
	Template specialization for vectorized functions applying to complex arguments. More...
struct	apply_scalar_unary< F, T, require_eigen_t< T > >
	Template specialization for vectorized functions applying to Eigen matrix arguments. More...
struct	apply_scalar_unary< F, T, require_floating_point_t< T > >
	Template specialization for vectorized functions applying to double arguments. More...
struct	apply_scalar_unary< F, T, require_integral_t< T > >
	Template specialization for vectorized functions applying to integer arguments. More...
struct	apply_scalar_unary< F, T, require_var_matrix_t< T > >
struct	apply_scalar_unary< F, var >
	Template specialization to var for vectorizing a unary scalar function. More...
struct	apply_vector_unary
struct	apply_vector_unary< T, require_eigen_t< T > >
	Base template class for vectorization of unary vector functions defined by applying a functor to a standard library vector, Eigen dense matrix expression template, or container of these. More...
struct	apply_vector_unary< T, require_std_vector_vt< is_container_or_var_matrix, T > >
	Specialisation for use with nested containers (std::vectors). More...
struct	apply_vector_unary< T, require_std_vector_vt< is_stan_scalar, T > >
	Specialisation for use with (non-nested) std::vectors. More...
struct	apply_vector_unary< T, require_var_matrix_t< T > >
	Specialisation for use with var_value<T> types where T inherits from EigenBase. More...
struct	arena_allocator
	std library compatible allocator that uses AD stack. More...
class	arena_matrix
	Equivalent to Eigen::Matrix, except that the data is stored on AD stack. More...
class	arena_matrix_cl
	A variant of matrix_cl that schedules its destructor to be called, so it can be used on the AD stack. More...
class	array_builder
	Structure for building up arrays in an expression (rather than in statements) using an argument-chaining add() method and a getter method array() to return the result. More...
class	as_column_vector_or_scalar_
	Represents as_column_vector_or_scalar of a row or column vector in kernel generator expressions. More...
class	asin_
struct	asin_fun
	Structure to wrap asin() so it can be vectorized. More...
class	asinh_
struct	asinh_fun
	Structure to wrap asinh() so it can be vectorized. More...
class	atan_
struct	atan_fun
	Structure to wrap atan() so it can be vectorized. More...
class	atanh_
struct	atanh_fun
	Structure to wrap atanh() so it can be vectorized. More...
struct	AutodiffStackSingleton
	This struct always provides access to the autodiff stack using the singleton pattern. More...
class	beta_
class	binary_operation
	Represents a binary operation in kernel generator expressions. More...
class	binomial_coefficient_log_
class	block_
	Represents submatrix block in kernel generator expressions. More...
class	broadcast_
	Represents a broadcasting operation in kernel generator expressions. More...
class	calc_if_
	Represents a calc_if in kernel generator expressions. More...
class	cast_
	Represents a typecast os scalar in kernel generator expressions. More...
class	cbrt_
struct	cbrt_fun
	Structure to wrap cbrt() so it can be vectorized. More...
class	ceil_
struct	ceil_fun
	Structure to wrap ceil() so it can be vectorized. More...
class	chainable_alloc
	A chainable_alloc is an object which is constructed and destructed normally but the memory lifespan is managed along with the arena allocator for the gradient calculation. More...
class	chainable_object
	chainable_object hold another object is useful for connecting the lifetime of a specific object to the chainable stack More...
class	check_cl_
	Represents a check in kernel generator expressions. More...
struct	child_type
	Primary template class for metaprogram to compute child type of T. More...
struct	child_type< T_struct< T_child > >
	Specialization for template classes / structs. More...
class	col_index
	Represents operation that determines column index. More...
class	colwise_max_
	Represents column wise max - reduction in kernel generator expressions. More...
class	colwise_min_
	Represents column wise min - reduction in kernel generator expressions. More...
class	colwise_prod_
	Represents column wise product - reduction in kernel generator expressions. More...
class	colwise_reduction
	Represents a column wise reduction in kernel generator expressions. More...
class	colwise_sum_
	Represents column wise sum - reduction in kernel generator expressions. More...
class	complex_base
	Base class for complex numbers. More...
struct	conjunction
	Extends std::true_type when instantiated with zero or more template parameters, all of which extend the std::true_type. More...
struct	conjunction< T, Ts... >
class	constant_
	Represents a matrix of single repeated value in kernel generator expressions. More...
class	copysign_
class	cos_
struct	cos_fun
	Structure to wrap cos() so it can be vectorized. More...
class	cosh_
struct	cosh_fun
	Structure to wrap cosh() so it can be vectorized. More...
struct	coupled_ode_system
struct	coupled_ode_system_impl
struct	coupled_ode_system_impl< false, F, T_y0, Args... >
	The coupled_ode_system_impl template specialization when the state or parameters are autodiff types. More...
struct	coupled_ode_system_impl< true, F, T_y0, Args... >
	The coupled_ode_system_impl for arithmetic arguments reduces to the regular ode system (there are no sensitivities) More...
class	cov_exp_quad_vari
class	cov_exp_quad_vari< T_x, double, T_l >
class	cvodes_integrator
	Integrator interface for CVODES' ODE solvers (Adams & BDF methods). More...
class	cvodes_integrator_adjoint_vari
	Integrator interface for CVODES' adjoint ODE solvers (Adams & BDF methods). More...
class	diagonal_
	Represents diagonal of a matrix (as column vector) in kernel generator expressions. More...
class	digamma_
struct	digamma_fun
	Structure to wrap digamma() so it can be vectorized. More...
struct	disjunction
	Extends std::false_type when instantiated with zero or more template parameters, all of which extend the std::false_type. More...
struct	disjunction< Cond, Conds... >
class	elt_divide_
class	elt_function_cl
	Represents an element-wise function in kernel generator expressions. More...
class	elt_modulo_
class	elt_multiply_
class	equals_
class	erf_
struct	erf_fun
	Structure to wrap erf() so it can be vectorized. More...
class	erfc_
struct	erfc_fun
	Structure to wrap the erfc() so that it can be vectorized. More...
class	exp2_
struct	exp2_fun
	Structure to wrap exp2() so it can be vectorized. More...
class	exp_
struct	exp_fun
	Structure to wrap exp() so that it can be vectorized. More...
class	expm1_
struct	expm1_fun
	Structure to wrap expm1() so that it can be vectorized. More...
class	expressions_cl
	Represents multiple expressions that will be calculated in same kernel. More...
class	fabs_
struct	fabs_fun
	Structure to wrap fabs() so that it can be vectorized. More...
class	fdim_
struct	FixedPointADJac
	Calculate Jacobian Jxy(Jacobian of unknown x w.r.t. More...
struct	FixedPointSolver
	Fixed point solver for problem of form. More...
struct	FixedPointSolver< KinsolFixedPointEnv< F >, fp_jac_type >
	Specialization for fixed point solver when using KINSOL. More...
class	floor_
struct	floor_fun
	Structure to wrap floor() so that it can be vectorized. More...
class	fmax_
class	fmin_
class	fmod_
struct	fvar
	This template class represents scalars used in forward-mode automatic differentiation, which consist of values and directional derivatives of the specified template type. More...
class	gevv_vvv_vari
class	greater_than_
class	greater_than_or_equal_
class	Holder
	A no-op Eigen operation. More...
class	holder_cl_
	Represents a no-op in kernel generator expressions. More...
struct	hybrj_functor_solver
	A functor with the required operators to call Eigen's algebraic solver. More...
class	hypot_
class	idas_forward_system
	IDAS DAE system with forward sensitivity calculation. More...
class	idas_integrator
	IDAS DAE integrator. More...
class	idas_system
	IDAS DAE system that contains information on residual equation functor, sensitivity residual equation functor, as well as initial conditions. More...
struct	include_summand
	Template metaprogram to calculate whether a summand needs to be included in a proportional (log) probability calculation. More...
struct	include_summand< propto, T >
	true if a term with the specified propto value and subterm types should be included in a proportionality calculation. More...
struct	index_type
	Primary template class for the metaprogram to compute the index type of a container. More...
struct	index_type< T, require_eigen_t< T > >
	Template metaprogram defining typedef for the type of index for an Eigen matrix, vector, or row vector. More...
struct	index_type< T, require_std_vector_t< T > >
	Template metaprogram class to compute the type of index for a standard vector. More...
struct	index_type< T, std::enable_if_t< std::is_pointer< T >::value > >
	Specialization of index_type for pointers. More...
class	indexing_
	Represents indexing of a matrix with two matrices of indices. More...
struct	inv_cloglog_fun
	Structure to wrap inv_cloglog() so that it can be vectorized. More...
struct	inv_fun
	Structure to wrap 1.0 / x so that it can be vectorized. More...
class	inv_logit_
struct	inv_logit_fun
	Structure to wrap inv_logit() so that it can be vectorized. More...
class	inv_Phi_
struct	inv_Phi_fun
	Structure to wrap inv_Phi() so it can be vectorized. More...
struct	inv_sqrt_fun
	Structure to wrap 1 / sqrt(x) so that it can be vectorized. More...
class	inv_square_
class	isfinite_
class	isinf_
class	isnan_
struct	kernel_parts
	Parts of an OpenCL kernel, generated by an expression. More...
class	kinsol_system_data
	KINSOL algebraic system data holder. More...
struct	KinsolFixedPointEnv
	KINSOL algebraic system data holder that handles construction & destruction of SUNDIALS data, as well as auxiliary data that will be used for functor evaluation. More...
class	lbeta_
class	ldexp_
class	LDLT_factor
	LDLT_factor is a structure that holds a matrix of type T and the LDLT of its values. More...
class	LDLT_factor< T, require_eigen_matrix_dynamic_vt< is_var, T > >
	An LDLT_factor of an Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic> with alloc_in_arena = True holds a copy of the input matrix and the LDLT of its values, with all member variable allocations are done in the arena. More...
class	LDLT_factor< T, require_var_matrix_t< T > >
	An LDLT_factor of a var_value<Eigen::MatrixXd> holds a copy of the input var_value and the LDLT of its values. More...
class	LDLT_factor< T, std::enable_if_t< bool_constant< is_eigen_matrix_dynamic< T >::value &&!is_var< scalar_type_t< T > >::value >::value > >
	An LDLT_factor is a structure that holds a matrix of type T and the LDLT of its values. More...
class	less_than_
class	less_than_or_equal_
class	lgamma_
struct	lgamma_fun
	Structure to wrap lgamma() so that it can be vectorized. More...
class	lmultiply_
class	load_
	Represents an access to a matrix_cl in kernel generator expressions. More...
class	log10_
struct	log10_fun
	Structure to wrap log10() so it can be vectorized. More...
class	log1m_
class	log1m_exp_
struct	log1m_exp_fun
	Structure to wrap log1m_exp() so it can be vectorized. More...
struct	log1m_fun
	Structure to wrap log1m() so it can be vectorized. More...
class	log1m_inv_logit_
struct	log1m_inv_logit_fun
	Structure to wrap log1m_inv_logit() so it can be vectorized. More...
class	log1p_
class	log1p_exp_
struct	log1p_exp_fun
	Structure to wrap log1p_exp() so that it can be vectorized. More...
struct	log1p_fun
	Structure to wrap log1p() so it can be vectorized. More...
class	log2_
struct	log2_fun
	Structure to wrap log2() so it can be vectorized. More...
class	log_
class	log_diff_exp_
struct	log_fun
	Structure to wrap log() so that it can be vectorized. More...
class	log_inv_logit_
class	log_inv_logit_diff_
struct	log_inv_logit_fun
	Structure to wrap log_inv_logit() so it can be vectorized. More...
class	logical_and_
class	logical_negation_
	Represents a logical negation in kernel generator expressions. More...
class	logical_or_
class	logit_
struct	logit_fun
	Structure to wrap logit() so it can be vectorized. More...
class	matrix_cl
	Represents an arithmetic matrix on the OpenCL device. More...
class	matrix_cl_base
	Non-templated base class for matrix_cl simplifies checking if something is matrix_cl. More...
class	matrix_exp_action_handler
	The implementation of the work by Awad H. More...
class	max_2d_
	Represents two dimensional max - reduction in kernel generator expressions. More...
struct	max_op
	Operation for max reduction. More...
class	min_2d_
	Represents two dimensional min - reduction in kernel generator expressions. More...
struct	min_op
	Operation for min reduction. More...
struct	mpi_cluster
	MPI cluster holds MPI resources and must be initialized only once in any MPI program. More...
struct	mpi_command
	A MPI command object is used to execute code on worker nodes. More...
struct	mpi_distributed_apply
	MPI command template which calls the static method distributed_apply of the given class F. More...
class	mpi_is_in_use
	Exception thrown whenever the MPI resource is busy. More...
class	mpi_parallel_call
	The MPI parallel call class manages the distributed evaluation of a collection of tasks following the map - reduce - combine pattern. More...
class	mpi_stop_listen
	Exception used to stop workers nodes from further listening to commands send from the root. More...
struct	mpi_stop_worker
	MPI command used to stop childs nodes from listening for further commands. More...
class	multiply_log_
class	name_generator
	Unique name generator for variables used in generated kernels. More...
class	nested_rev_autodiff
	A class following the RAII idiom to start and recover nested autodiff scopes. More...
struct	nlo_functor
	A structure which gets passed to Eigen's dogleg algebraic solver. More...
class	not_equals_
class	op_ddv_vari
class	op_dv_vari
class	op_dvd_vari
class	op_dvv_vari
class	op_matrix_vari
class	op_v_vari
class	op_vd_vari
class	op_vdd_vari
class	op_vdv_vari
class	op_vector_vari
class	op_vv_vari
class	op_vvd_vari
class	op_vvv_vari
class	opencl_code_
	Represents custom code in kernel generator expressions. More...
class	opencl_code_output
	Represents output variable of custom code in kernel generator expressions. More...
class	opencl_context
	The API to access the methods and values in opencl_context_base. More...
class	opencl_context_base
	The opencl_context_base class represents an OpenCL context in the standard Meyers singleton design pattern. More...
class	operands_and_partials
	This template builds partial derivatives with respect to a set of operands. More...
class	operands_and_partials< Op1, Op2, Op3, Op4, Op5, fvar< Dx > >
	This class builds partial derivatives with respect to a set of operands. More...
class	operands_and_partials< Op1, Op2, Op3, Op4, Op5, var >
	This class builds partial derivatives with respect to a set of operands. More...
class	operation_cl
	Base for all kernel generator operations. More...
class	operation_cl_lhs
	Base for all kernel generator operations that can be used on left hand side of an expression. More...
class	optional_broadcast_
	Represents an optional broadcasting operation in kernel generator expressions. More...
struct	pass_type
struct	pass_type< double >
struct	pass_type< int >
class	Phi_
class	Phi_approx_
struct	Phi_approx_fun
	Structure to wrap Phi_approx() so it can be vectorized. More...
struct	Phi_fun
	Structure to wrap Phi() so it can be vectorized. More...
class	pinned_matrix
	Equivalent to Eigen::Matrix, except that the data is stored in (hopefully pinned) memory, allocated by OpenCL driver. More...
class	pow_
class	precomp_vv_vari
class	precomp_vvv_vari
class	precomputed_gradients_vari_template
	A variable implementation taking a sequence of operands and partial derivatives with respect to the operands. More...
class	prod_2d_
	Represents two dimensional product - reduction in kernel generator expressions. More...
struct	prod_op
	Operation for product reduction. More...
class	profile
	Profiles C++ lines where the object is in scope. More...
class	profile_info
	Class used for storing profiling information. More...
struct	promote_elements
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< Eigen::Matrix< T, R, C >, Eigen::Matrix< S, R, C > >
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< Eigen::Matrix< T, R, C >, Eigen::Matrix< T, R, C > >
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< std::vector< T >, std::vector< S > >
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< std::vector< T >, std::vector< T > >
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< T, T >
	Struct with static function for elementwise type promotion. More...
struct	promote_scalar_struct
	General struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_struct< T, S, require_eigen_t< S > >
	Struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_struct< T, std::vector< S > >
	Struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_struct< T, T >
	Struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_type
	Template metaprogram to calculate a type for converting a convertible type. More...
struct	promote_scalar_type< T, S, require_all_t< is_var< T >, is_var< S >, is_eigen< value_type_t< S > > > >
	Specialization for var_value when the type to convert to is a var_value. More...
struct	promote_scalar_type< T, S, require_all_t< std::is_arithmetic< T >, is_var< S >, is_eigen< value_type_t< S > > > >
	Specialization for var_value when the type to convert to is arithmetic. More...
struct	promote_scalar_type< T, S, require_eigen_t< S > >
	Template metaprogram to calculate a type for a matrix, vector, row vector or Eigen::Array whose underlying scalar is converted from the second template parameter type to the first. More...
struct	promote_scalar_type< T, std::vector< S > >
	Template metaprogram to calculate a type for a container whose underlying scalar is converted from the second template parameter type to the first. More...
class	read_fvar_functor
	Functor for extracting the values and tangents from a matrix of fvar. More...
class	reduction_2d
	Represents a two dimensional reduction in kernel generator expressions. More...
struct	ref_type_for_opencl
	Determines appropriate type for assigning expression of given type to, so that the resulting type has directly accessible contiguous colum-major data, which is needed to copy to OpenCL device for construction of matrix_cl. More...
struct	ref_type_for_opencl< T, require_arena_matrix_t< T > >
struct	ref_type_for_opencl< T, require_not_eigen_t< T > >
class	results_cl
	Represents results that will be calculated in same kernel. More...
class	round_
struct	round_fun
	Structure to wrap round() so it can be vectorized. More...
class	row_index
	Represents operation that determines row index. More...
class	rowwise_max_
	Represents rowwise max reduction in kernel generator expressions. More...
class	rowwise_min_
	Represents rowwise min reduction in kernel generator expressions. More...
class	rowwise_prod_
	Represents rowwise product reduction in kernel generator expressions. More...
class	rowwise_reduction
	Represents a rowwise reduction in kernel generator expressions. More...
class	rowwise_sum_
	Represents rowwise sum reduction in kernel generator expressions. More...
class	rsqrt_
class	scalar_
	Represents a scalar in kernel generator expressions. More...
class	ScopedChainableStack
	The AD tape of reverse mode AD is by default stored globally within the process (or thread). More...
class	select_
	Represents a selection operation in kernel generator expressions. More...
class	seq_view
class	seq_view< double, std::vector< int > >
class	seq_view< T, Eigen::Matrix< S, 1, Eigen::Dynamic > >
class	seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, 1 > >
class	seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, Eigen::Dynamic > >
class	seq_view< T, std::vector< S > >
class	seq_view< T, std::vector< std::vector< T > > >
class	seq_view< T, std::vector< T > >
struct	sign_fun
	Structure to wrap sign() so it can be vectorized. More...
class	sin_
struct	sin_fun
	Structure to wrap sin() so it can be vectorized. More...
class	sinh_
struct	sinh_fun
	Structure to wrap sinh() so that it can be vectorized. More...
class	sqrt_
struct	sqrt_fun
	Structure to wrap sqrt() so that it can be vectorized. More...
class	square_
struct	square_fun
	Structure to wrap square() so that it can be vectorized. More...
class	stack_alloc
	An instance of this class provides a memory pool through which blocks of raw memory may be allocated and then collected simultaneously. More...
struct	step_fun
	Structure to wrap step() so it can be vectorized. More...
struct	store_type
struct	store_type< double >
struct	store_type< int >
class	stored_gradient_vari
	A var implementation that stores the daughter variable implementation pointers and the partial derivative with respect to the result explicitly in arrays constructed on the autodiff memory stack. More...
class	subtraction_
class	subtraction_operator_
class	sum_2d_
	Represents two dimensional sum - reduction in kernel generator expressions. More...
struct	sum_op
	Operation for sum reduction. More...
class	tan_
struct	tan_fun
	Structure to wrap tan() so that it can be vectorized. More...
class	tanh_
struct	tanh_fun
	Structure to wrap tanh() so that it can be vectorized. More...
class	tgamma_
struct	tgamma_fun
	Structure to wrap tgamma() so that it can be vectorized. More...
class	transpose_
	Represents a transpose in kernel generator expressions. More...
class	trigamma_
struct	trigamma_fun
	Structure to wrap trigamma() so it can be vectorized. More...
class	trunc_
struct	trunc_fun
	Structure to wrap trunc() so it can be vectorized. More...
class	unary_minus_
	Represents an unary minus operation in kernel generator expressions. More...
class	unary_operation_cl
	Represents a unary operation in kernel generator expressions. More...
class	unsafe_chainable_object
	unsafe_chainable_object hold another object and is useful for connecting the lifetime of a specific object to the chainable stack. More...
class	val_adj_functor
	Functor for extracting the values and adjoints from a matrix of var or vari. More...
class	var_value
class	var_value< T, internal::require_matrix_var_value< T > >
	Independent (input) and dependent (output) variables for gradients. More...
class	var_value< T, require_floating_point_t< T > >
	Independent (input) and dependent (output) variables for gradients. More...
class	vari_base
	Abstract base class that all vari_value and it's derived classes inherit. More...
class	vari_cl_base
class	vari_value
class	vari_value< T, require_all_t< is_plain_type< T >, is_eigen_dense_base< T > > >
	The variable implementation for Eigen dense matrix types. More...
class	vari_value< T, require_eigen_sparse_base_t< T > >
	The variable implementation for Eigen sparse matrix types. More...
class	vari_value< T, require_matrix_cl_t< T > >
	The variable implementation for matrix_cl. More...
class	vari_value< T, require_t< std::is_floating_point< T > > >
	The variable implementation for floating point types. More...
class	vari_view
	A vari_view is used to read from a slice of a vari_value with an inner eigen type. More...
class	vari_view< T, require_all_t< is_eigen< T >, bool_constant<!is_plain_type< T >::value > > >
class	vari_view< T, require_kernel_expression_lhs_t< T > >
class	vari_view_eigen
	This struct is follows the CRTP for methods common to vari_view<> and vari_value<Matrix>. More...
class	vi_adj_functor
	Functor for extracting the varis and adjoints from a matrix of var. More...
class	vi_val_adj_functor
	Functor for extracting the vari*, values, and adjoints from a matrix of var. More...
class	vi_val_functor
	Functor for extracting the varis and values from a matrix of var. More...
class	welford_covar_estimator
class	welford_var_estimator

Typedefs
using	size_type = Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index
	Type for sizes and indexes in an Eigen matrix with double elements. More...
using	matrix_fd = Eigen::Matrix< fvar< double >, Eigen::Dynamic, Eigen::Dynamic >
using	matrix_ffd = Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, Eigen::Dynamic >
using	vector_fd = Eigen::Matrix< fvar< double >, Eigen::Dynamic, 1 >
using	vector_ffd = Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, 1 >
using	row_vector_fd = Eigen::Matrix< fvar< double >, 1, Eigen::Dynamic >
using	row_vector_ffd = Eigen::Matrix< fvar< fvar< double > >, 1, Eigen::Dynamic >
using	matrix_fv = Eigen::Matrix< fvar< var >, Eigen::Dynamic, Eigen::Dynamic >
using	matrix_ffv = Eigen::Matrix< fvar< fvar< var >>, Eigen::Dynamic, Eigen::Dynamic >
using	vector_fv = Eigen::Matrix< fvar< var >, Eigen::Dynamic, 1 >
using	vector_ffv = Eigen::Matrix< fvar< fvar< var >>, Eigen::Dynamic, 1 >
using	row_vector_fv = Eigen::Matrix< fvar< var >, 1, Eigen::Dynamic >
using	row_vector_ffv = Eigen::Matrix< fvar< fvar< var >>, 1, Eigen::Dynamic >
template<typename T >
using	as_operation_cl_t = std::conditional_t< std::is_lvalue_reference< T >::value, decltype(as_operation_cl(std::declval< T >())), std::remove_reference_t< decltype(as_operation_cl(std::declval< T >()))>>
	Type that results when converting any valid kernel generator expression into operation. More...
template<typename T >
using	is_without_output = internal::is_without_output_impl< std::decay_t< T >>
template<typename T >
using	is_colwise_reduction = internal::is_colwise_reduction_impl< std::decay_t< T >>
	Check whether a kernel generator expression is a colwise reduction. More...
template<typename... Types>
using	common_scalar_t = typename std::common_type_t< typename std::remove_reference_t< Types >::Scalar... >
	Wrapper for std::common_type_t More...
template<typename T >
using	is_reduction_2d = internal::is_reduction_2d_impl< std::decay_t< T >>
	Check whether a kernel generator expression is a colwise reduction. More...
template<typename T >
using	ref_type_for_opencl_t = typename ref_type_for_opencl< T >::type
template<int B = 0>
using	boost_policy_t = boost::math::policies::policy< boost::math::policies::overflow_error< boost::math::policies::errno_on_error >, boost::math::policies::pole_error< boost::math::policies::errno_on_error >, boost::math::policies::promote_double< false >, boost::math::policies::digits2< B >>
	Boost policy that overrides the defaults to match the built-in C++ standard library functions. More...
using	matrix_d = Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >
	Type for matrix of double values. More...
using	vector_d = Eigen::Matrix< double, Eigen::Dynamic, 1 >
	Type for (column) vector of double values. More...
using	row_vector_d = Eigen::Matrix< double, 1, Eigen::Dynamic >
	Type for (row) vector of double values. More...
template<typename T >
using	index_type_t = typename index_type< T >::type
	Type for sizes and indexes in an Eigen matrix with double elements. More...
template<typename T , typename S >
using	promote_scalar_t = typename promote_scalar_type< std::decay_t< T >, std::decay_t< S >>::type
using	ChainableStack = AutodiffStackSingleton< vari_base, chainable_alloc >
using	precomputed_gradients_vari = precomputed_gradients_vari_template< std::tuple<>, std::tuple<>>
using	profile_key = std::pair< std::string, std::thread::id >
using	profile_map = std::map< profile_key, profile_info >
using	matrix_v = Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic >
	The type of a matrix holding var values. More...
using	vector_v = Eigen::Matrix< var, Eigen::Dynamic, 1 >
	The type of a (column) vector holding var values. More...
using	row_vector_v = Eigen::Matrix< var, 1, Eigen::Dynamic >
	The type of a row vector holding var values. More...
using	matrix_vi = Eigen::Matrix< vari *, Eigen::Dynamic, Eigen::Dynamic >
	The type of a matrix holding vari* values. More...
using	vector_vi = Eigen::Matrix< vari *, Eigen::Dynamic, 1 >
	The type of a (column) vector holding vari* values. More...
using	row_vector_vi = Eigen::Matrix< vari *, 1, Eigen::Dynamic >
	The type of a row vector holding vari* values. More...
using	var = var_value< double >
using	vari = vari_value< double >

Enumerations
enum	matrix_cl_view { matrix_cl_view::Diagonal = 0, matrix_cl_view::Lower = 1, matrix_cl_view::Upper = 2, matrix_cl_view::Entire = 3 }

Functions
template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, const fvar< T > &x2)
	Return the sum of the specified forward mode addends. More...
template<typename T >
fvar< T >	operator+ (double x1, const fvar< T > &x2)
	Return the sum of the specified double and forward mode addends. More...
template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, double x2)
	Return the sum of the specified forward mode and double addends. More...
template<typename T >
fvar< T >	operator/ (const fvar< T > &x1, const fvar< T > &x2)
	Return the result of dividing the first argument by the second. More...
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
fvar< T >	operator/ (const fvar< T > &x1, U x2)
	Return the result of dividing the first argument by the second. More...
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
fvar< T >	operator/ (U x1, const fvar< T > &x2)
	Return the result of dividing the first argument by the second. More...
template<typename T >
std::complex< fvar< T > >	operator/ (const std::complex< fvar< T >> &x1, const std::complex< fvar< T >> &x2)
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
std::complex< fvar< T > >	operator/ (const std::complex< fvar< T >> &x1, const std::complex< U > &x2)
template<typename T >
std::complex< fvar< T > >	operator/ (const std::complex< fvar< T >> &x1, const fvar< T > &x2)
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
std::complex< fvar< T > >	operator/ (const std::complex< fvar< T >> &x1, U x2)
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
std::complex< fvar< T > >	operator/ (const std::complex< U > &x1, const std::complex< fvar< T >> &x2)
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
std::complex< fvar< T > >	operator/ (const std::complex< U > &x1, const fvar< T > &x2)
template<typename T >
std::complex< fvar< T > >	operator/ (const fvar< T > &x1, const std::complex< fvar< T >> &x2)
template<typename T , typename U , typename = std::enable_if_t<std::is_arithmetic<U>::value>>
std::complex< fvar< T > >	operator/ (const fvar< T > &x1, const std::complex< U > &x2)
template<typename T , typename U , require_arithmetic_t< U > * = nullptr>
std::complex< fvar< T > >	operator/ (U x1, const std::complex< fvar< T >> &x2)
template<typename T >
bool	operator== (const fvar< T > &x, const fvar< T > &y)
	Return true if the specified variables have equal values as defined by ==. More...
template<typename T >
bool	operator== (const fvar< T > &x, double y)
	Return true if the the first variable has a value equal to the second argument as defined by by ==. More...
template<typename T >
bool	operator== (double x, const fvar< T > &y)
	Return true if the the first argument is equal to the value of the second argument as defined by by ==. More...
template<typename T >
bool	operator> (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator> (const fvar< T > &x, double y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator> (double x, const fvar< T > &y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator>= (const fvar< T > &x, const fvar< T > &y)
	Return true if the value of the first argument is greater than or equal to that of the second as defined by >=. More...
template<typename T >
bool	operator>= (const fvar< T > &x, double y)
	Return true if the value of the first argument has a value greater than or equal to the second argument as defined by >=. More...
template<typename T >
bool	operator>= (double x, const fvar< T > &y)
	Return true if the first argument is greater than or equal to the value of the second argument as defined by >=. More...
template<typename T >
bool	operator< (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a value less than the value of the second argument as defined by <. More...
template<typename T >
bool	operator< (double x, const fvar< T > &y)
	Return true if the first argument is less than the value of the second argument as defined by <. More...
template<typename T >
bool	operator< (const fvar< T > &x, double y)
	Return true if the first argument has a value less than the second argument as defined by <. More...
template<typename T >
bool	operator<= (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a value less than or equal to the value of the second argument as defined by <=. More...
template<typename T >
bool	operator<= (const fvar< T > &x, double y)
	Return true if the first argument has a value less than or equal to the second argument as defined by <=. More...
template<typename T >
bool	operator<= (double x, const fvar< T > &y)
	Return true if the first argument is less than or equal to the second argument's value as defined by <=. More...
template<typename T >
bool	operator&& (const fvar< T > &x, const fvar< T > &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (const fvar< T > &x, double y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (double x, const fvar< T > &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator|| (const fvar< T > &x, const fvar< T > &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (const fvar< T > &x, double y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (double x, const fvar< T > &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
fvar< T >	operator* (const fvar< T > &x, const fvar< T > &y)
	Return the product of the two arguments. More...
template<typename T >
fvar< T >	operator* (double x, const fvar< T > &y)
	Return the product of the two arguments. More...
template<typename T >
fvar< T >	operator* (const fvar< T > &x, double y)
	Return the product of the two arguments. More...
template<typename T >
bool	operator!= (const fvar< T > &x, const fvar< T > &y)
	Return true if the value of the two arguments are not equal as defined by !=. More...
template<typename T >
bool	operator!= (const fvar< T > &x, double y)
	Return true if the value of the first argument is not equal to the second argument as defined by !=. More...
template<typename T >
bool	operator!= (double x, const fvar< T > &y)
	Return true if the first argument is not equal to the value of the second argument as defined by !=. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x1, const fvar< T > &x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (double x1, const fvar< T > &x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x1, double x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x)
	Return the negation of the specified argument. More...
template<typename T >
bool	operator! (const fvar< T > &x)
	Return the negation of the value of the argument as defined by !. More...
template<typename T >
fvar< T >	operator+ (const fvar< T > &x)
	Returns the argument. More...
template<typename T >
fvar< T >	abs (const fvar< T > &x)
template<typename T >
fvar< T >	abs (const std::complex< fvar< T >> &z)
	Return the absolute value of the complex argument. More...
template<typename T >
fvar< T >	acos (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	acos (const std::complex< fvar< T >> &x)
	Return the arc cosine of the complex argument. More...
template<typename T >
fvar< T >	acosh (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	acosh (const std::complex< fvar< T >> &z)
	Return the hyperbolic arc cosine of the complex argument. More...
template<typename T >
fvar< T >	arg (const std::complex< fvar< T >> &z)
	Return the phase angle of the complex argument. More...
template<typename T >
fvar< T >	asin (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	asin (const std::complex< fvar< T >> &z)
	Return the arc sine of the complex argument. More...
template<typename T >
fvar< T >	asinh (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	asinh (const std::complex< fvar< T >> &z)
	Return the hyperbolic arcsine of the complex argument. More...
template<typename T >
fvar< T >	atan (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	atan (const std::complex< fvar< T >> &z)
	Return the arc tangent of the complex argument. More...
template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	atan2 (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	atanh (const fvar< T > &x)
	Return inverse hyperbolic tangent of specified value. More...
template<typename T >
std::complex< fvar< T > >	atanh (const std::complex< fvar< T >> &z)
	Return the hyperbolic arc tangent of the complex argument. More...
template<typename T >
fvar< T >	bessel_first_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	bessel_second_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	beta (const fvar< T > &x1, const fvar< T > &x2)
	Return fvar with the beta function applied to the specified arguments and its gradient. More...
template<typename T >
fvar< T >	beta (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	beta (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	binary_log_loss (int y, const fvar< T > &y_hat)
template<typename T >
fvar< T >	cbrt (const fvar< T > &x)
	Return cube root of specified argument. More...
template<typename T >
fvar< T >	ceil (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	conj (const std::complex< fvar< T >> &z)
	Return the phase angle of the complex argument. More...
template<typename T >
fvar< T >	cos (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	cos (const std::complex< fvar< T >> &z)
	Return the cosine of the complex argument. More...
template<typename T >
fvar< T >	cosh (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	cosh (const std::complex< fvar< T >> &z)
	Return the hyperbolic cosine of the complex argument. More...
template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>
value_type_t< EigMat >	determinant (const EigMat &m)
template<typename T >
fvar< T >	digamma (const fvar< T > &x)
	Return the derivative of the log gamma function at the specified argument. More...
template<typename T >
fvar< T >	erf (const fvar< T > &x)
template<typename T >
fvar< T >	erfc (const fvar< T > &x)
template<typename T >
fvar< T >	exp (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	exp (const std::complex< fvar< T >> &z)
	Return the natural exponentiation (base e) of the specified complex number. More...
template<typename T >
fvar< T >	exp2 (const fvar< T > &x)
template<typename T >
fvar< T >	expm1 (const fvar< T > &x)
template<typename T >
fvar< T >	fabs (const fvar< T > &x)
template<typename T >
fvar< T >	falling_factorial (const fvar< T > &x, int n)
	Return autodiff variable with the gradient and result of the falling factorial function applied to the inputs. More...
template<typename T >
fvar< T >	fdim (const fvar< T > &x, const fvar< T > &y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	fdim (const fvar< T > &x, double y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	fdim (double x, const fvar< T > &y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	floor (const fvar< T > &x)
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	The fused multiply-add operation (C99). More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const T1 &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const fvar< T1 > &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const T1 &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const fvar< T1 > &x1, const T2 &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>
fvar< return_type_t< T1, T2, T3 > >	fma (const T1 &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T >
fvar< T >	fmax (const fvar< T > &x1, const fvar< T > &x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmax (double x1, const fvar< T > &x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmax (const fvar< T > &x1, double x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmin (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmin (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmin (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	fmod (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmod (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	fmod (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	gamma_p (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	gamma_q (double x1, const fvar< T > &x2)
template<typename T >
void	grad_inc_beta (fvar< T > &g1, fvar< T > &g2, fvar< T > a, fvar< T > b, fvar< T > z)
	Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments. More...
template<typename T >
fvar< T >	hypot (const fvar< T > &x1, const fvar< T > &x2)
	Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	hypot (const fvar< T > &x1, double x2)
	Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	hypot (double x1, const fvar< T > &x2)
	Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, const fvar< T > &b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (double a, const fvar< T > &b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, double b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, const fvar< T > &b, double x)
template<typename T >
fvar< T >	inc_beta (double a, double b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, double b, double x)
template<typename T >
fvar< T >	inc_beta (double a, const fvar< T > &b, double x)
template<typename T >
fvar< T >	inv (const fvar< T > &x)
template<typename T >
fvar< T >	inv_cloglog (const fvar< T > &x)
template<typename T >
fvar< T >	inv_logit (const fvar< T > &x)
	Returns the inverse logit function applied to the argument. More...
template<typename T >
fvar< T >	inv_Phi (const fvar< T > &p)
template<typename T >
fvar< T >	inv_sqrt (const fvar< T > &x)
template<typename T >
fvar< T >	inv_square (const fvar< T > &x)
template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime >	inverse (const EigMat &m)
	Forward mode specialization of calculating the inverse of the matrix. More...
template<typename T >
int	is_inf (const fvar< T > &x)
	Returns 1 if the input's value is infinite and 0 otherwise. More...
template<typename T , require_fvar_t< T > * = nullptr>
bool	is_nan (T &&x)
	Returns 1 if the input's value is NaN and 0 otherwise. More...
template<typename T >
fvar< T >	lambert_w0 (const fvar< T > &x)
template<typename T >
fvar< T >	lambert_wm1 (const fvar< T > &x)
template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lbeta (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	ldexp (const fvar< T > &a, int b)
	Returns the product of a (the significand) times 2 to power b (the exponent). More...
template<typename T >
fvar< T >	lgamma (const fvar< T > &x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
template<typename T >
fvar< return_type_t< T, int > >	lmgamma (int x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lmultiply (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lmultiply (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lmultiply (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	log (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	log (const std::complex< fvar< T >> &z)
	Return the natural logarithm (base e) of the specified complex argument. More...
template<typename T >
fvar< T >	log10 (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	log10 (const std::complex< fvar< T >> &z)
	Return the base 10 logarithm of the specified complex number. More...
template<typename T >
fvar< T >	log1m (const fvar< T > &x)
template<typename T >
fvar< T >	log1m_exp (const fvar< T > &x)
	Return the natural logarithm of one minus the exponentiation of the specified argument. More...
template<typename T >
fvar< T >	log1m_inv_logit (const fvar< T > &x)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
template<typename T >
fvar< T >	log1p (const fvar< T > &x)
template<typename T >
fvar< T >	log1p_exp (const fvar< T > &x)
template<typename T >
fvar< T >	log2 (const fvar< T > &x)
	Return the base two logarithm of the specified argument. More...
template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>
value_type_t< EigMat >	log_determinant (const EigMat &m)
	Returns the log absolute determinant of the specified square matrix. More...
template<typename T >
fvar< T >	log_diff_exp (const fvar< T > &x1, const fvar< T > &x2)
template<typename T1 , typename T2 , require_arithmetic_t< T1 > * = nullptr>
fvar< T2 >	log_diff_exp (const T1 &x1, const fvar< T2 > &x2)
template<typename T1 , typename T2 , require_arithmetic_t< T2 > * = nullptr>
fvar< T1 >	log_diff_exp (const fvar< T1 > &x1, const T2 &x2)
template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, const fvar< T > &n)
template<typename T >
fvar< T >	log_falling_factorial (double x, const fvar< T > &n)
template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, double n)
template<typename T >
fvar< T >	log_inv_logit (const fvar< T > &x)
template<typename T >
fvar< T >	log_inv_logit_diff (const fvar< T > &x, const fvar< T > &y)
	Returns fvar with the natural logarithm of the difference of the inverse logits of the specified arguments and its gradients. More...
template<typename T >
fvar< T >	log_inv_logit_diff (const fvar< T > &x, double y)
template<typename T >
fvar< T >	log_inv_logit_diff (double x, const fvar< T > &y)
template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>
void	log_mix_partial_helper (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2, promote_args_t< T_theta, T_lambda1, T_lambda2 >(&partials_array)[N])
template<typename T >
fvar< T >	log_mix (const fvar< T > &theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
	Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>
fvar< T >	log_mix (const fvar< T > &theta, const fvar< T > &lambda1, P lambda2)
template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>
fvar< T >	log_mix (const fvar< T > &theta, P lambda1, const fvar< T > &lambda2)
template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>
fvar< T >	log_mix (P theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>
fvar< T >	log_mix (const fvar< T > &theta, P1 lambda1, P2 lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>
fvar< T >	log_mix (P1 theta, const fvar< T > &lambda1, P2 lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>
fvar< T >	log_mix (P1 theta, P2 lambda1, const fvar< T > &lambda2)
template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, const fvar< T > &n)
template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, double n)
template<typename T >
fvar< T >	log_rising_factorial (double x, const fvar< T > &n)
template<typename T , require_vector_st< is_fvar, T > * = nullptr>
auto	log_softmax (const T &x)
	Return the log softmax of the specified vector or container of vectors. More...
template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	log_sum_exp (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, double x2)
template<typename T , require_container_st< is_fvar, T > * = nullptr>
auto	log_sum_exp (const T &x)
	Return the log of the sum of the exponentiated values of the specified matrix of values. More...
template<typename T >
fvar< T >	logit (const fvar< T > &x)
template<typename T1 , typename T2 , require_all_eigen_vt< is_fvar, T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T1 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left (const T1 &A, const T2 &b)
template<typename T1 , typename T2 , require_eigen_vt< std::is_arithmetic, T1 > * = nullptr, require_eigen_vt< is_fvar, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T2 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left (const T1 &A, const T2 &b)
template<typename T , typename EigMat , require_eigen_vt< std::is_arithmetic, T > * = nullptr, require_eigen_vt< is_fvar, EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, EigMat::ColsAtCompileTime >	mdivide_left_ldlt (LDLT_factor< T > &A, const EigMat &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<typename T1 , typename T2 , require_all_eigen_vt< is_fvar, T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T1 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri_low (const T1 &A, const T2 &b)
template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_vt_same< double, T1 > * = nullptr, require_eigen_vt< is_fvar, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T2 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri_low (const T1 &A, const T2 &b)
template<typename T1 , typename T2 , require_eigen_vt< is_fvar, T1 > * = nullptr, require_eigen_t< T2 > * = nullptr, require_vt_same< double, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T1 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri_low (const T1 &A, const T2 &b)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< is_fvar, EigMat1, EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< value_type_t< EigMat1 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right (const EigMat1 &A, const EigMat2 &b)
template<typename EigMat1 , typename EigMat2 , require_eigen_vt< std::is_arithmetic, EigMat1 > * = nullptr, require_eigen_vt< is_fvar, EigMat2 > * = nullptr>
Eigen::Matrix< value_type_t< EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right (const EigMat1 &A, const EigMat2 &b)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< is_fvar, EigMat1, EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< value_type_t< EigMat1 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right_tri_low (const EigMat1 &A, const EigMat2 &b)
template<typename EigMat1 , typename EigMat2 , require_eigen_vt< std::is_arithmetic, EigMat1 > * = nullptr, require_eigen_vt< is_fvar, EigMat2 > * = nullptr>
Eigen::Matrix< value_type_t< EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right_tri_low (const EigMat1 &A, const EigMat2 &b)
template<typename T >
fvar< T >	modified_bessel_first_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	modified_bessel_second_kind (int v, const fvar< T > &z)
template<typename Mat1 , typename Mat2 , require_all_eigen_vt< is_fvar, Mat1, Mat2 > * = nullptr, require_vt_same< Mat1, Mat2 > * = nullptr, require_not_eigen_row_and_col_t< Mat1, Mat2 > * = nullptr>
auto	multiply (const Mat1 &m1, const Mat2 &m2)
	Return the product of the specified matrices. More...
template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	multiply_log (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, double x2)
template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::RowsAtCompileTime >	multiply_lower_tri_self_transpose (const EigMat &m)
template<typename T >
fvar< T >	norm (const std::complex< fvar< T >> &z)
	Return the squared magnitude of the complex argument. More...
template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, const fvar< T > &x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	owens_t (double x1, const fvar< T > &x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, double x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	Phi (const fvar< T > &x)
template<typename T >
fvar< T >	Phi_approx (const fvar< T > &x)
	Return an approximation of the unit normal cumulative distribution function (CDF). More...
template<typename T >
std::complex< fvar< T > >	polar (const fvar< T > &r, const fvar< T > &theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T , typename U >
std::complex< fvar< T > >	polar (const fvar< T > &r, U theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T , typename U >
std::complex< fvar< T > >	polar (U r, const fvar< T > &theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T >
fvar< T >	pow (const fvar< T > &x1, const fvar< T > &x2)
template<typename T , typename U , typename = require_arithmetic_t<U>>
fvar< T >	pow (U x1, const fvar< T > &x2)
template<typename T , typename U , typename = require_arithmetic_t<U>>
fvar< T >	pow (const fvar< T > &x1, U x2)
template<typename V >
std::complex< fvar< V > >	pow (const std::complex< fvar< V >> &x, const std::complex< fvar< V >> &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V , typename T , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (const std::complex< fvar< V >> &x, const std::complex< T > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V >
std::complex< fvar< V > >	pow (const std::complex< fvar< V >> &x, const fvar< V > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V , typename T , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (const std::complex< fvar< V >> &x, const T &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V , typename T , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (const std::complex< T > &x, const std::complex< fvar< V >> &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V , typename T , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (const std::complex< T > &x, const fvar< V > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V >
std::complex< fvar< V > >	pow (const fvar< V > &x, const std::complex< fvar< V >> &y)
	Return the first argument raised to the power of the second argument. More...
template<typename V , typename T , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (const fvar< V > &x, const std::complex< T > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename V , typename = require_arithmetic_t<T>>
std::complex< fvar< V > >	pow (T x, const std::complex< fvar< V >> &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T >
std::complex< fvar< T > >	pow (const std::complex< fvar< T >> &x, int y)
	Return the first argument raised to the power of the second argument. More...
template<typename T >
double	primitive_value (const fvar< T > &v)
	Return the primitive value of the specified forward-mode autodiff variable. More...
template<typename T >
std::complex< fvar< T > >	proj (const std::complex< fvar< T >> &z)
	Return the projection of the complex argument onto the Riemann sphere. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>
promote_scalar_t< return_type_t< EigMat1, EigMat2 >, EigMat2 >	quad_form (const EigMat1 &A, const EigMat2 &B)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_fvar< EigMat, ColVec > * = nullptr>
return_type_t< EigMat, ColVec >	quad_form (const EigMat &A, const ColVec &B)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>
promote_scalar_t< return_type_t< EigMat1, EigMat2 >, EigMat2 >	quad_form_sym (const EigMat1 &A, const EigMat2 &B)
	Return the quadratic form \( B^T A B \) of a symmetric matrix. More...
template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_fvar< EigMat, ColVec > * = nullptr>
return_type_t< EigMat, ColVec >	quad_form_sym (const EigMat &A, const ColVec &B)
	Return the quadratic form \( B^T A B \) of a symmetric matrix. More...
template<typename EigFvar , typename EigOut >
void	read_fvar (const EigFvar &FvarMat, EigOut &ValMat, EigOut &DMat)
	Function applying the read_fvar_functor to extract the values and tangets of a given fvar matrix into separate matrices. More...
template<typename T >
fvar< T >	rising_factorial (const fvar< T > &x, int n)
	Return autodiff variable with the gradient and result of the rising factorial function applied to the inputs. More...
template<typename T >
fvar< T >	round (const fvar< T > &x)
	Return the closest integer to the specified argument, with halfway cases rounded away from zero. More...
template<typename T >
fvar< T >	sin (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	sin (const std::complex< fvar< T >> &z)
	Return the sine of the complex argument. More...
template<typename T >
fvar< T >	sinh (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	sinh (const std::complex< fvar< T >> &z)
	Return the hyperbolic sine of the complex argument. More...
template<typename ColVec , require_eigen_col_vector_vt< is_fvar, ColVec > * = nullptr>
auto	softmax (const ColVec &alpha)
template<typename T >
fvar< T >	sqrt (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	sqrt (const std::complex< fvar< T >> &z)
	Return the square root of the complex argument. More...
template<typename T >
fvar< T >	square (const fvar< T > &x)
template<typename T >
fvar< T >	sum (const std::vector< fvar< T >> &m)
	Return the sum of the entries of the specified standard vector. More...
template<typename T , require_eigen_vt< is_fvar, T > * = nullptr>
value_type_t< T >	sum (const T &m)
	Return the sum of the entries of the specified matrix. More...
template<typename T >
fvar< T >	tan (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	tan (const std::complex< fvar< T >> &z)
	Return the tangent of the complex argument. More...
template<typename T >
fvar< T >	tanh (const fvar< T > &x)
template<typename T >
std::complex< fvar< T > >	tanh (const std::complex< fvar< T >> &z)
	Return the hyperbolic tangent of the complex argument. More...
template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::RowsAtCompileTime >	tcrossprod (const EigMat &m)
template<typename T >
fvar< T >	tgamma (const fvar< T > &x)
	Return the result of applying the gamma function to the specified argument. More...
template<typename T , require_stan_scalar_t< T > * = nullptr, require_not_fvar_t< T > * = nullptr>
fvar< T >	to_fvar (const T &x)
template<typename T , require_fvar_t< scalar_type_t< T >> * = nullptr>
T &&	to_fvar (T &&x)
	Specialization of to_fvar for [containers of] fvars. More...
template<typename T >
std::vector< fvar< T > >	to_fvar (const std::vector< T > &v)
template<typename T >
std::vector< fvar< T > >	to_fvar (const std::vector< T > &v, const std::vector< T > &d)
template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_fvar, T > * = nullptr>
promote_scalar_t< fvar< value_type_t< T > >, T >	to_fvar (const T &m)
template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>
promote_scalar_t< fvar< value_type_t< T1 > >, T1 >	to_fvar (const T1 &val, const T2 &deriv)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>
return_type_t< EigMat1, EigMat2 >	trace_quad_form (const EigMat1 &A, const EigMat2 &B)
template<typename T >
fvar< T >	trigamma (const fvar< T > &u)
	Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument). More...
template<typename T >
fvar< T >	trunc (const fvar< T > &x)
	Return the nearest integral value that is not larger in magnitude than the specified argument. More...
template<typename EigMat , require_eigen_col_vector_vt< is_fvar, EigMat > * = nullptr>
auto	unit_vector_constrain (const EigMat &y)
template<typename EigMat , typename T , require_eigen_vt< is_fvar, EigMat > * = nullptr, require_stan_scalar_t< T > * = nullptr>
auto	unit_vector_constrain (const EigMat &y, T &lp)
template<typename T >
T	value_of (const fvar< T > &v)
	Return the value of the specified variable. More...
template<typename T >
double	value_of_rec (const fvar< T > &v)
	Return the value of the specified variable. More...
template<typename T , typename F >
void	gradient (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument. More...
template<typename T , typename F >
void	hessian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &H)
	Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in time O(N^3) time and O(N^2) space. More...
template<typename T , typename F >
void	jacobian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, Eigen::Matrix< T, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &J)
template<typename T >
bool	is_aligned (T *ptr, unsigned int bytes_aligned)
	Return true if the specified pointer is aligned on the number of bytes. More...
template<typename T , typename F >
void	derivative (const F &f, const T &x, T &fx, T &dfx_dx)
	Return the derivative of the specified univariate function at the specified argument. More...
template<typename F >
void	finite_diff_grad_hessian (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::MatrixXd &hess, std::vector< Eigen::MatrixXd > &grad_hess_fx, double epsilon=1e-04)
	Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference. More...
template<typename F >
void	finite_diff_grad_hessian_auto (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::MatrixXd &hess, std::vector< Eigen::MatrixXd > &grad_hess_fx)
	Calculate the value, Hessian, and the gradient of the Hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference. More...
template<typename F >
void	grad_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H, std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &grad_H)
	Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument. More...
template<typename F >
void	grad_tr_mat_times_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &M, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_tr_MH)
template<typename T1 , typename T2 , typename F >
void	gradient_dot_vector (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &v, T1 &fx, T1 &grad_fx_dot_v)
template<typename F >
void	hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H)
	Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space. More...
template<typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &v, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &Hv)
template<typename T , typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &Hv)
template<typename T , typename F >
void	partial_derivative (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int n, T &fx, T &dfx_dxn)
	Return the partial derivative of the specified multivariate function at the specified argument. More...
template<typename T , require_st_arithmetic< T > * = nullptr>
matrix_cl< scalar_type_t< T > >	to_matrix_cl (T &&src)
	Copies the source Eigen matrix, std::vector or scalar to the destination matrix that is stored on the OpenCL device. More...
template<typename T_ret , typename T , require_eigen_t< T_ret > * = nullptr, require_matrix_cl_t< T > * = nullptr, require_st_same< T_ret, T > * = nullptr>
auto	from_matrix_cl (const T &src)
	Copies the source matrix that is stored on the OpenCL device to the destination Eigen matrix. More...
template<typename T_ret , typename T , require_all_kernel_expressions_t< T > * = nullptr, require_not_matrix_cl_t< T > * = nullptr>
auto	from_matrix_cl (const T &src)
	Copies result of a kernel generator expression to the specified destination type. More...
template<typename T_dst , typename T , require_arithmetic_t< T > * = nullptr, require_same_t< T_dst, T > * = nullptr>
T_dst	from_matrix_cl (const matrix_cl< T > &src)
	Copy A 1 by 1 source matrix from the Device to the host. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	from_matrix_cl (const T &src)
	Copies the source kernel generator expression or matrix that is stored on the OpenCL device to the destination Eigen matrix. More...
template<typename T , require_matrix_cl_t< T > * = nullptr>
auto	packed_copy (const T &src)
	Packs the square flat triangular matrix on the OpenCL device and copies it to the std::vector. More...
template<matrix_cl_view matrix_view, typename Vec , typename Vec_scalar = scalar_type_t<Vec>, require_vector_vt< std::is_arithmetic, Vec > * = nullptr>
matrix_cl< Vec_scalar >	packed_copy (Vec &&src, int rows)
	Copies the packed triangular matrix from the source std::vector to an OpenCL buffer and unpacks it to a flat matrix on the OpenCL device. More...
template<typename T , require_matrix_cl_t< T > * = nullptr>
plain_type_t< T >	copy_cl (const T &src)
	Copies the source matrix to the destination matrix. More...
template<typename T >
void	check_mat_not_size_one (const char *function, const char *name, const matrix_cl< T > &A)
	Check if the matrix_cl has a single element. More...
void	check_opencl_error (const char *function, const cl::Error &e)
	Throws the domain error with specifying the OpenCL error that occurred. More...
template<typename T , typename = require_arithmetic_t<T>>
void	check_symmetric (const char *function, const char *name, const matrix_cl< T > &y)
	Check if the matrix_cl is symmetric. More...
template<typename T , require_matrix_cl_t< T > * = nullptr>
void	check_triangular (const char *function, const char *name, const T &A)
	Check if the matrix_cl is either upper triangular or lower triangular. More...
void	indexing_rev (matrix_cl< double > &adj, const matrix_cl< int > &idx, const matrix_cl< double > &res)
	Performs reverse pass for indexing operation on the OpenCL device. More...
template<typename Ta , typename Tb , typename = require_all_kernel_expressions_and_none_scalar_t<Ta, Tb>>
auto	append_row (Ta &&a, Tb &&b)
	Stack the rows of the first argument on top of the second argument. More...
template<typename Ta , typename Tb , typename = require_all_kernel_expressions_and_none_scalar_t<Ta, Tb>>
auto	append_col (Ta &&a, Tb &&b)
	Stack the cols of the arguments. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	as_column_vector_or_scalar (T &&a)
	as_column_vector_or_scalar of a kernel generator expression. More...
template<typename T_operation , typename = std::enable_if_t<std::is_base_of< operation_cl_base, std::remove_reference_t<T_operation>>::value>>
T_operation &&	as_operation_cl (T_operation &&a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_scalar , typename = require_arithmetic_t<T_scalar>, require_not_same_t< T_scalar, bool > * = nullptr>
scalar_< T_scalar >	as_operation_cl (const T_scalar a)
	Converts any valid kernel generator expression into an operation. More...
scalar_< char >	as_operation_cl (const bool a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_matrix_cl , typename = require_any_t<is_matrix_cl<T_matrix_cl>, is_arena_matrix_cl<T_matrix_cl>>>
load_< T_matrix_cl >	as_operation_cl (T_matrix_cl &&a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
addition_operator_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator+ (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
addition_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	add (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
subtraction_operator_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator- (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
subtraction_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	subtract (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
elt_multiply_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	elt_multiply (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
elt_divide_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	elt_divide (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
elt_modulo_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator% (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
less_than_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator< (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
less_than_or_equal_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator<= (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
greater_than_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator> (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
greater_than_or_equal_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator>= (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
equals_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator== (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
not_equals_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator!= (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
logical_or_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator|| (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_any_not_arithmetic_t< T_a, T_b > * = nullptr>
logical_and_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator&& (T_a &&a, T_b &&b)
template<typename T_a , typename T_b , typename = require_arithmetic_t<T_a>, typename = require_all_kernel_expressions_t<T_b>>
elt_multiply_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator* (T_a a, T_b &&b)
	Multiplication of a scalar and a kernel generator expression. More...
template<typename T_a , typename T_b , typename = require_all_kernel_expressions_t<T_a>, typename = require_arithmetic_t<T_b>>
elt_multiply_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator* (T_a &&a, const T_b b)
	Multiplication of a kernel generator expression and a scalar. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	block_zero_based (T &&a, int start_row, int start_col, int rows, int cols)
	Block of a kernel generator expression. More...
template<bool Colwise, bool Rowwise, typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	broadcast (T &&a)
	Broadcast an expression in specified dimension(s). More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_broadcast (T &&a)
	Broadcast an expression in rowwise dimmension. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	colwise_broadcast (T &&a)
	Broadcast an expression in colwise dimmension. More...
template<bool Do_Calculate, typename T , require_all_kernel_expressions_t< T > * = nullptr, std::enable_if_t< Do_Calculate > * = nullptr>
calc_if_< true, as_operation_cl_t< T > >	calc_if (T &&a)
template<bool Do_Calculate, typename T , std::enable_if_t<!Do_Calculate > * = nullptr>
calc_if_< false, scalar_< double > >	calc_if (T &&a)
template<typename Scalar , typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	cast (T &&a)
	Typecast a kernel generator expression scalar. More...
template<typename Scalar , typename T , require_stan_scalar_t< T > * = nullptr>
Scalar	cast (T a)
	Typecast a scalar. More...
template<typename T , typename = require_all_kernel_expressions_t<T>>
auto	check_cl (const char *function, const char *var_name, T &&y, const char *must_be)
	Constructs a check on opencl matrix or expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	colwise_sum (T &&a)
	Column wise sum - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	colwise_prod (T &&a)
	Column wise product - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	colwise_max (T &&a)
	Column wise max - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	colwise_min (T &&a)
	Column wise min - reduction of a kernel generator expression. More...
template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>
T1	operator+= (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>
T1	operator-= (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>
T1	operator*= (T1 &&a, T2 &&b)
template<typename T , typename = require_arithmetic_t<T>>
auto	constant (const T a, int rows, int cols)
	Matrix of repeated values in kernel generator expressions. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	diagonal (T &&a)
	Diagonal of a kernel generator expression. More...
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
rsqrt_< as_operation_cl_t< T > >	rsqrt (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
sqrt_< as_operation_cl_t< T > >	sqrt (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
cbrt_< as_operation_cl_t< T > >	cbrt (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
exp_< as_operation_cl_t< T > >	exp (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
exp2_< as_operation_cl_t< T > >	exp2 (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
expm1_< as_operation_cl_t< T > >	expm1 (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log_< as_operation_cl_t< T > >	log (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log2_< as_operation_cl_t< T > >	log2 (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log10_< as_operation_cl_t< T > >	log10 (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log1p_< as_operation_cl_t< T > >	log1p (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
sin_< as_operation_cl_t< T > >	sin (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
sinh_< as_operation_cl_t< T > >	sinh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
cos_< as_operation_cl_t< T > >	cos (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
cosh_< as_operation_cl_t< T > >	cosh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
tan_< as_operation_cl_t< T > >	tan (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
tanh_< as_operation_cl_t< T > >	tanh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
asin_< as_operation_cl_t< T > >	asin (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
asinh_< as_operation_cl_t< T > >	asinh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
acos_< as_operation_cl_t< T > >	acos (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
acosh_< as_operation_cl_t< T > >	acosh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
atan_< as_operation_cl_t< T > >	atan (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
atanh_< as_operation_cl_t< T > >	atanh (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
tgamma_< as_operation_cl_t< T > >	tgamma (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
lgamma_< as_operation_cl_t< T > >	lgamma (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
erf_< as_operation_cl_t< T > >	erf (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
erfc_< as_operation_cl_t< T > >	erfc (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
floor_< as_operation_cl_t< T > >	floor (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
round_< as_operation_cl_t< T > >	round (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
ceil_< as_operation_cl_t< T > >	ceil (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
fabs_< as_operation_cl_t< T > >	fabs (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
trunc_< as_operation_cl_t< T > >	trunc (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
digamma_< as_operation_cl_t< T > >	digamma (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log1m_< as_operation_cl_t< T > >	log1m (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log_inv_logit_< as_operation_cl_t< T > >	log_inv_logit (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log1m_exp_< as_operation_cl_t< T > >	log1m_exp (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log1p_exp_< as_operation_cl_t< T > >	log1p_exp (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
inv_square_< as_operation_cl_t< T > >	inv_square (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
inv_logit_< as_operation_cl_t< T > >	inv_logit (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
logit_< as_operation_cl_t< T > >	logit (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
Phi_< as_operation_cl_t< T > >	Phi (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
Phi_approx_< as_operation_cl_t< T > >	Phi_approx (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
inv_Phi_< as_operation_cl_t< T > >	inv_Phi (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
log1m_inv_logit_< as_operation_cl_t< T > >	log1m_inv_logit (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
trigamma_< as_operation_cl_t< T > >	trigamma (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
square_< as_operation_cl_t< T > >	square (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
isfinite_< as_operation_cl_t< T > >	isfinite (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
isinf_< as_operation_cl_t< T > >	isinf (T &&a)
template<typename T , typename Cond = require_all_kernel_expressions_and_none_scalar_t<T>>
isnan_< as_operation_cl_t< T > >	isnan (T &&a)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
fdim_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	fdim (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
fmax_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	fmax (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
fmin_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	fmin (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
fmod_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	fmod (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
hypot_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	hypot (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
ldexp_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	ldexp (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
pow_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	pow (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
copysign_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	copysign (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
beta_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	beta (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
binomial_coefficient_log_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	binomial_coefficient_log (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
lbeta_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	lbeta (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
log_inv_logit_diff_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	log_inv_logit_diff (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
log_diff_exp_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	log_diff_exp (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
multiply_log_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	multiply_log (T1 &&a, T2 &&b)
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr, require_any_not_stan_scalar_t< T1, T2 > * = nullptr>
lmultiply_< as_operation_cl_t< T1 >, as_operation_cl_t< T2 > >	lmultiply (T1 &&a, T2 &&b)
template<typename T , typename... Ptrs, require_all_kernel_expressions_t< T, Ptrs... > * = nullptr>
auto	holder_cl (T &&a, Ptrs *...ptrs)
	Constructs a no-op operation that also holds pointer to some other expressions, allocated on heap. More...
template<typename T , typename... Args, require_all_kernel_expressions_t< decltype(std::declval< T >()(std::declval< Args & >()...)), Args... > * = nullptr>
auto	make_holder_cl (const T &func, Args &&...args)
	Constructs an expression from given arguments using given functor. More...
template<typename T_mat , typename T_row_index , typename T_col_index , require_all_kernel_expressions_t< T_mat, T_row_index, T_col_index > * = nullptr>
auto	indexing (T_mat &&mat, T_row_index &&row_index, T_col_index &&col_index)
	Index a kernel generator expression using two expressions for indices. More...
template<typename T_matrix , typename T_vector , typename = require_all_kernel_expressions_t<T_matrix, T_vector>>
auto	matrix_vector_multiply (T_matrix &&matrix, T_vector &&vector)
	Multiplies a matrix and a vector on an OpenCL device. More...
template<typename... T_expressions>
expressions_cl< T_expressions... >	expressions (T_expressions &&...expressions)
	Deduces types for constructing expressions_cl object. More...
template<typename... T_results>
results_cl< T_results... >	results (T_results &&...results)
	Deduces types for constructing results_cl object. More...
template<const char * Code, typename... T_arguments, require_all_kernel_expressions_t< T_arguments... > * = nullptr>
auto	opencl_code (std::tuple< typename std::pair< const char *, T_arguments >::first_type... > names, T_arguments &&...arguments)
	Custom code in kernel generator expressions. More...
template<bool Colwise, bool Rowwise, typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
optional_broadcast_< as_operation_cl_t< T >, Colwise, Rowwise >	optional_broadcast (T &&a)
	Broadcast an expression in specified dimension(s) if the size along that dimension equals 1. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_optional_broadcast (T &&a)
	Broadcast an expression in rowwise dimmension if the number of columns equals to 1. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	colwise_optional_broadcast (T &&a)
	Broadcast an expression in colwise dimmension if the number of rows equals to 1. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	sum_2d (T &&a)
	Two dimensional sum - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	prod_2d (T &&a)
	Two dimensional product - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	max_2d (T &&a)
	Two dimensional max - reduction of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	min_2d (T &&a)
	Two dimensional min - reduction of a kernel generator expression. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_sum (T &&a)
	Rowwise sum reduction of a kernel generator expression. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_prod (T &&a)
	Rowwise product reduction of a kernel generator expression. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_max (T &&a)
	Rowwise max reduction of a kernel generator expression. More...
template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>
auto	rowwise_min (T &&a)
	Min reduction of a kernel generator expression. More...
template<typename T_condition , typename T_then , typename T_else , require_all_kernel_expressions_t< T_condition, T_then, T_else > * = nullptr, require_any_not_arithmetic_t< T_condition, T_then, T_else > * = nullptr>
select_< as_operation_cl_t< T_condition >, as_operation_cl_t< T_then >, as_operation_cl_t< T_else > >	select (T_condition &&condition, T_then &&then, T_else &&els)
	Selection operation on kernel generator expressions. More...
template<typename T_then , typename T_else , require_all_arithmetic_t< T_then, T_else > * = nullptr>
std::common_type_t< T_then, T_else >	select (bool condition, T_then then, T_else els)
	Scalar overload of the selection operation. More...
template<typename Arg , typename = require_all_kernel_expressions_and_none_scalar_t<Arg>>
auto	transpose (Arg &&a)
	Transposes a kernel generator expression. More...
template<typename T >
std::string	type_str ()
	Determines a string name of a type. More...
template<>
std::string	type_str< double > ()
template<>
std::string	type_str< int > ()
template<>
std::string	type_str< char > ()
template<>
std::string	type_str< bool > ()
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
logical_negation_< as_operation_cl_t< T > >	operator! (T &&a)
	Logical negation of a kernel generator expression. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
unary_minus_< as_operation_cl_t< T > >	operator- (T &&a)
	Unary minus of a kernel generator expression. More...
const matrix_cl_view	either (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in any of the input views. More...
const matrix_cl_view	both (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in both input views. More...
bool	contains_nonzero (const matrix_cl_view view, const matrix_cl_view part)
	Check whether a view contains certain nonzero part. More...
const matrix_cl_view	transpose (const matrix_cl_view view)
	Transposes a view - swaps lower and upper parts. More...
const matrix_cl_view	invert (const matrix_cl_view view)
	Inverts a view. More...
matrix_cl_view	from_eigen_uplo_type (Eigen::UpLoType eigen_type)
	Creates a view from Eigen::UpLoType. More...
int	either (int left_view, int right_view)
	Determines which parts are nonzero in any of the input views. More...
int	both (int left_view, int right_view)
	Determines which parts are nonzero in both input views. More...
bool	contains_nonzero (int view, int part)
	Check whether a view contains certain nonzero part. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	multiply_transpose (const matrix_cl< T > &A)
	Computes the product of a square OpenCL matrix with its transpose. More...
template<typename T_m , typename T_a , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr, require_all_kernel_expressions_t< T_a > * = nullptr>
auto	add_diag (T_m &&mat, T_a &&to_add)
	Returns a Matrix with values added along the main diagonal. More...
template<typename T_x , typename T_y , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr>
auto	append_array (T_x &&x, T_y &&y)
	Return the concatenation of two specified vectors in the order of the arguments. More...
template<typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	bernoulli_cdf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the CDF of the Bernoulli distribution. More...
template<typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	bernoulli_lccdf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the log CCDF of the Bernoulli distribution. More...
template<typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	bernoulli_lcdf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the log CDF of the Bernoulli distribution. More...
template<bool propto, typename T_x_cl , typename T_y_cl , typename T_alpha_cl , typename T_beta_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl >	bernoulli_logit_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<bool propto, typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	bernoulli_logit_lpmf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the log PMF of the logit-parametrized Bernoulli distribution. More...
template<bool propto, typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	bernoulli_lpmf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the log PMF of the Bernoulli distribution. More...
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_size1_cl , typename T_size2_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_size1_cl, T_size2_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_size1_cl, T_size2_cl > * = nullptr>
return_type_t< T_n_cl, T_size1_cl, T_size2_cl >	beta_binomial_lpmf (const T_n_cl &n, const T_N_cl N, const T_size1_cl &alpha, const T_size2_cl &beta)
	Returns the log PMF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<bool propto, typename T_y_cl , typename T_scale_succ_cl , typename T_scale_fail_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl >	beta_lpdf (const T_y_cl &y, const T_scale_succ_cl &alpha, const T_scale_fail_cl &beta)
	The log of the beta density for the specified scalar(s) given the specified sample stan::math::size(s). More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_prec_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_prec_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_prec_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_prec_cl >	beta_proportion_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_prec_cl &kappa)
	The log of the beta density for specified y, location, and precision: beta_proportion_lpdf(y | mu, kappa) = beta_lpdf(y | mu * kappa, (1 - mu) * kappa). More...
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr, require_any_nonscalar_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	binomial_logit_lpmf (const T_n_cl &n, const T_N_cl N, const T_prob_cl &alpha)
	Binomial log PMF in logit parametrization. More...
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr, require_any_nonscalar_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	binomial_lpmf (const T_n_cl &n, const T_N_cl N, const T_prob_cl &theta)
	Returns the log PMF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>
auto	block (T_x &&x, size_t i, size_t j, size_t nrows, size_t ncols)
	Return a nrows x ncols submatrix starting at (i-1, j-1). More...
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , require_all_prim_or_rev_kernel_expression_t< T_y, T_x, T_alpha, T_beta > * = nullptr>
return_type_t< T_x, T_alpha, T_beta >	categorical_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with categorical distribution and logit (softmax) link function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	cauchy_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	cauchy_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	cauchy_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	cauchy_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<bool propto, typename T_y_cl , typename T_dof_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl >	chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
matrix_cl< double >	cholesky_decompose (const matrix_cl< double > &A)
	Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix on the OpenCL device. More...
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	col (T_x &&x, size_t j)
	Return the specified column of the specified kernel generator expression using start-at-1 indexing. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
int	cols (const T_x &x)
	Returns the number of columns in the specified kernel generator expression. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>
auto	columns_dot_product (const T_a &a, const T_b &b)
	Returns the dot product of columns of the specified matrices. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	columns_dot_self (const T &a)
	Returns the dot product of each column of a matrix with itself. More...
template<typename T_A , typename = require_all_kernel_expressions_and_none_scalar_t<T_A>>
matrix_cl< typename std::decay_t< T_A >::Scalar >	crossprod (T_A &&A)
	Returns the result of pre-multiplying a matrix by its own transpose. More...
template<typename T_vec , require_all_kernel_expressions_and_none_scalar_t< T_vec > * = nullptr>
auto	cumulative_sum (T_vec &&v)
	Return the cumulative sum of the specified vector. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	diag_matrix (T_x &&x)
	Return a square diagonal matrix with the specified vector of coefficients as the diagonal values. More...
template<typename T1 , typename T2 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr>
auto	diag_post_multiply (const T1 &m1, const T2 &m2)
	Return the product of a matrix and the diagonal matrix formed from the vector or row_vector. More...
template<typename T1 , typename T2 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr>
auto	diag_pre_multiply (const T1 &m1, const T2 &m2)
	Return the product of the diagonal matrix formed from the vector or row_vector and a matrix. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
void	dims (const T_x &x, std::vector< int > &result)
	matrix_cl overload of the dims helper function in prim/fun/dims.hpp. More...
template<bool propto, typename T_prob_cl , typename T_prior_size_cl , require_all_prim_or_rev_kernel_expression_t< T_prob_cl, T_prior_size_cl > * = nullptr, require_any_not_stan_scalar_t< T_prob_cl, T_prior_size_cl > * = nullptr>
return_type_t< T_prob_cl, T_prior_size_cl >	dirichlet_lpdf (const T_prob_cl &theta, const T_prior_size_cl &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr>
auto	distance (const T_a &a, const T_b &b)
	Returns the distance between the specified vectors. More...
template<typename T_a , typename = require_all_kernel_expressions_and_none_scalar_t<T_a>>
auto	divide (T_a &&a, double d)
	Returns the elementwise division of the kernel generator expression. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	divide_columns (matrix_cl< T1 > &A, const matrix_cl< T2 > &B)
	Divides each column of a matrix by a vector. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	divide_columns (matrix_cl< T1 > &A, const T2 &divisor)
	Divides each column of a matrix by a scalar. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>
auto	dot_product (const T_a &a, const T_b &b)
	Returns the dot product of the specified vectors. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	dot_self (const T &a)
	Returns squared norm of a vector or matrix. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	double_exponential_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	double_exponential_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log complementary cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	double_exponential_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log cumulative density function. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	double_exponential_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log probability density function. More...
matrix_cl< double >	eigenvalues_sym (const matrix_cl< double > &m)
matrix_cl< double >	eigenvectors_sym (const matrix_cl< double > &m)
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	exp_mod_normal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the double exponential cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	exp_mod_normal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the exp mod normal log complementary cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	exp_mod_normal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the exp mod normal log cumulative density function. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	exp_mod_normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the log PMF of the exp mod normal distribution. More...
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	exponential_cdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the exponential cumulative distribution function for the given y and beta. More...
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	exponential_lccdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the log exponential cumulative distribution function for the given y and beta. More...
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	exponential_lcdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the log exponential cumulative distribution function for the given y and beta. More...
template<bool propto, typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	exponential_lpdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	The log of an exponential density for y with the specified inverse scale parameter. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	frechet_cdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	frechet_lccdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	frechet_lcdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	frechet_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	The log of the frechet density for the specified scalar(s) given the specified sample stan::math::size(s). More...
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_inv_scale_cl >	gamma_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_inv_scale_cl &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters. More...
template<typename T_x , typename T_sigma , require_all_prim_or_rev_kernel_expression_t< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
auto	gp_dot_prod_cov (const T_x &x, const T_sigma sigma)
	Dot product kernel on the GPU. More...
template<typename T_x , typename T_y , typename T_sigma , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
auto	gp_dot_prod_cov (const T_x &x, const T_y &y, const T_sigma sigma)
	Dot product kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename = require_all_arithmetic_t<T1, T2, T3>>
matrix_cl< return_type_t< T1, T2, T3 > >	gp_exp_quad_cov (const matrix_cl< T1 > &x, const T2 sigma, const T3 length_scale)
	Squared exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , typename = require_all_arithmetic_t<T1, T2, T3, T4>>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	gp_exp_quad_cov (const matrix_cl< T1 > &x, const matrix_cl< T2 > &y, const T3 sigma, const T4 length_scale)
	Squared exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	gp_exponential_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	gp_exponential_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	gp_matern32_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern 3/2 kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	gp_matern32_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern 3/2 kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	gp_matern52_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern 5/2 kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	gp_matern52_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern 5/2 kernel on the GPU. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	gumbel_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the gumbel cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	gumbel_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	gumbel_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log complementary cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	gumbel_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log probability density for the given location and scale. More...
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	head (T_x &&x, size_t n)
	Return the specified number of elements as a vector or row vector (same as input) from the front of the specified vector or row vector. More...
template<typename T_x , require_matrix_cl_t< T_x > * = nullptr>
auto	identity_matrix (int K)
	Return a square identity matrix. More...
template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>
auto	inv (T_x &&x)
	Return the elementwise 1.0 / x of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<bool propto, typename T_y_cl , typename T_dof_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl >	inv_chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>
auto	inv_cloglog (T_x &&x)
	The inverse complementary log-log function. More...
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	inv_gamma_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>
auto	inv_sqrt (T_x &&x)
	Return the elementwise 1 / sqrt(x) of the specified kernel generator expression. More...
template<typename T , typename L , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L > * = nullptr>
auto	lb_constrain (T &&x, L &&lb)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T , typename L , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L > * = nullptr>
auto	lb_constrain (const T &x, const L &lb, return_type_t< T, L > &lp)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T_theta_cl , typename T_lambda_cl , require_all_prim_or_rev_kernel_expression_t< T_theta_cl, T_lambda_cl > * = nullptr>
auto	log_mix (const T_theta_cl &theta, const T_lambda_cl &lambda)
	Return the log mixture density with specified mixing proportions and log densities. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
matrix_cl< double >	log_softmax (const T &a)
	Return log of the softmax of the specified vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
double	log_sum_exp (const T &a)
	Return the log of the sum of the exponentiated values of the specified matrix of values. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	logistic_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	logistic_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	logistic_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	logistic_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of a logistic density for y with the specified location and scale parameters. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	lognormal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the loghormal cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	lognormal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the lognormal log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	lognormal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the lognormal log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	lognormal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the lognormal density for the specified scalar(s) given the specified sample stan::math::size(s). More...
template<typename T , typename L , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L, U > * = nullptr>
matrix_cl< double >	lub_constrain (const T &x, const L &lb, const U &ub)
	Return the lower and upper-bounded matrix derived by transforming the specified free matrix given the specified lower and upper bounds. More...
template<typename T , typename L , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L, U > * = nullptr>
auto	lub_constrain (const T &x, const L &lb, const U &ub, return_type_t< T, L, U > &lp)
	Return the lower and upper-bounded matrix derived by transforming the specified free matrix given the specified lower and upper bounds. More...
template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>
plain_type_t< T_m >	matrix_power (T_m &&M, const int n)
	Returns the nth power of the specific matrix. More...
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr>
matrix_cl< double >	mdivide_left_tri_low (const T1 &A, const T2 &b)
	Returns the solution of the system Ax=b when A is lower triangular. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
matrix_cl< double >	mdivide_left_tri_low (const T &A)
	Returns the solution of the system Ax=b when A is triangular and b=I. More...
template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr>
matrix_cl< double >	mdivide_right_tri_low (const T2 &b, const T1 &A)
	Returns the solution of the system Ax=b where A is a lower triangular matrix. More...
template<typename T , require_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
scalar_type_t< T >	mean (const T &m)
	Returns the sample mean (i.e., average) of the coefficients in the specified std vector, vector, row vector, or matrix. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_covar_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_covar_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_covar_cl >	multi_normal_cholesky_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_covar_cl &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
template<typename T1 , typename T2 , typename = require_all_kernel_expressions_and_none_scalar_t<T1, T2>>
matrix_cl< return_type_t< T1, T2 > >	multiply (const T1 &A, const T2 &B)
	Computes the product of the specified matrices with the option of specifying the triangularity of either input matrices. More...
template<typename T_a , typename T_b , typename = require_all_kernel_expressions_and_none_scalar_t<T_a, T_b>>
matrix_cl< return_type_t< T_a, T_b > >	operator* (const T_a &a, const T_b &b)
	Matrix multiplication of two kernel generator expressions. More...
template<typename T_a , typename T_b , require_stan_scalar_t< T_a > * = nullptr, require_all_kernel_expressions_and_none_scalar_t< T_b > * = nullptr, require_all_not_var_t< T_a, T_b > * = nullptr>
matrix_cl< return_type_t< T_a, T_b > >	multiply (const T_a &a, const T_b &b)
	Matrix multiplication of a scalar and a kernel generator expressions. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	multiply_lower_tri_self_transpose (T_x &&x)
	Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose. More...
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , typename T_phi_cl , require_all_prim_or_rev_kernel_expression_t< T_x_cl, T_y_cl, T_alpha_cl, T_beta_cl, T_phi_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl, T_phi_cl >	neg_binomial_2_log_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta, const T_phi_cl &phi)
	Returns the log PMF of the Generalized Linear Model (GLM) with Negative-Binomial-2 distribution and log link function. More...
template<bool propto, typename T_n_cl , typename T_log_location_cl , typename T_precision_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_log_location_cl, T_precision_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_log_location_cl, T_precision_cl > * = nullptr>
return_type_t< T_n_cl, T_log_location_cl, T_precision_cl >	neg_binomial_2_log_lpmf (const T_n_cl &n, const T_log_location_cl &eta, const T_precision_cl &phi)
	The log of the log transformed negative binomial density for the specified scalars given the specified mean(s) and deviation(s). More...
template<bool propto, typename T_n_cl , typename T_location_cl , typename T_precision_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_location_cl, T_precision_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_location_cl, T_precision_cl > * = nullptr>
return_type_t< T_n_cl, T_location_cl, T_precision_cl >	neg_binomial_2_lpmf (const T_n_cl &n, const T_location_cl &mu, const T_precision_cl &phi)
	The log of the negative binomial density for the specified scalars given the specified mean(s) and deviation(s). More...
template<bool propto, typename T_n_cl , typename T_shape_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_shape_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_shape_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_n_cl, T_shape_cl, T_inv_scale_cl >	neg_binomial_lpmf (const T_n_cl &n, const T_shape_cl &alpha, const T_inv_scale_cl &beta)
	The log of the negative binomial density for the specified scalars given the specified mean(s) and deviation(s). More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	normal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , typename T_sigma_cl , require_all_prim_or_rev_kernel_expression_t< T_x_cl, T_y_cl, T_alpha_cl, T_beta_cl, T_sigma_cl > * = nullptr>
return_type_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl, T_sigma_cl >	normal_id_glm_lpdf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta, const T_sigma_cl &sigma)
	Returns the log PDF of the Generalized Linear Model (GLM) with Normal distribution and id link function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	normal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	normal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal log complementary cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T , require_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
size_t	num_elements (const T &m)
	Returns the number of the elements of a matrix_cl or var_value<matrix_cl<T>>. More...
template<typename T , typename M , typename S , require_all_kernel_expressions_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr>
auto	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<typename T , typename M , typename S , require_all_kernel_expressions_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr>
auto	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma, double &lp)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<bool propto, typename T_y , typename T_x , typename T_beta , typename T_cuts , require_all_prim_or_rev_kernel_expression_t< T_y, T_x, T_beta, T_cuts > * = nullptr>
return_type_t< T_x, T_beta, T_cuts >	ordered_logistic_glm_lpmf (const T_y &y, const T_x &x, const T_beta &beta, const T_cuts &cuts)
	Returns the log PMF of the ordinal regression Generalized Linear Model (GLM). More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_cuts_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_cuts_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_cuts_cl >	ordered_logistic_lpmf (const T_y_cl &y, const T_loc_cl &lambda, const T_cuts_cl &cuts)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and specified cutpoints in an ordered logistic model. More...
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	pareto_cdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function. More...
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	pareto_lccdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function. More...
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	pareto_lcdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function. More...
template<bool propto, typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	pareto_lpdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	pareto_type_2_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	pareto_type_2_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 log complementaty cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	pareto_type_2_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 log cumulative density function. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	pareto_type_2_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the log PMF of the Pareto type 2 distribution. More...
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl >	poisson_log_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Poisson distribution and log link function. More...
template<bool propto, typename T_n_cl , typename T_log_rate_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_log_rate_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_log_rate_cl > * = nullptr>
return_type_t< T_log_rate_cl >	poisson_log_lpmf (const T_n_cl &n, const T_log_rate_cl &alpha)
	Returns the log PMF of the Poisson log distribution. More...
template<bool propto, typename T_n_cl , typename T_rate_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_rate_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_rate_cl > * = nullptr>
return_type_t< T_rate_cl >	poisson_lpmf (const T_n_cl &n, const T_rate_cl &lambda)
	Returns the log PMF of the Poisson distribution. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
value_type_t< T >	prod (const T &m)
	Calculates product of given kernel generator expression elements. More...
template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>
matrix_cl< double >	qr_Q (T_m &&m)
	Returns the orthogonal factor of the fat QR decomposition. More...
template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>
matrix_cl< double >	qr_R (T_m &&m)
	Returns the orthogonal factor of the fat QR decomposition. More...
template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>
matrix_cl< double >	qr_thin_Q (T_m &&m)
	Returns the orthogonal factor of the fat QR decomposition. More...
template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>
matrix_cl< double >	qr_thin_R (T_m &&m)
	Returns the orthogonal factor of the thin QR decomposition. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
int	rank (const T &v, int s)
	Return the number of components of v less than v[s]. More...
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	rayleigh_cdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	rayleigh_lccdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	rayleigh_lcdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	rayleigh_lpdf (const T_y_cl &y, const T_scale_cl &sigma)
	The log of an Rayleigh density for y with the specified scale parameter. More...
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T >>>> * = nullptr>
auto	rep_array (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing an array by replicating the input value. More...
template<typename T , require_matrix_cl_t< T > * = nullptr>
auto	rep_matrix (const value_type_t< T > &x, int n, int m)
	Creates a matrix_cl by replicating the given value of arithmetic type. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
auto	rep_matrix (const matrix_cl< T > &x, int m)
	Creates a matrix_cl by replicating the input vector or row_vector. More...
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T >>>> * = nullptr>
auto	rep_row_vector (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing a row vector by replicating the input value. More...
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T >>>> * = nullptr>
auto	rep_vector (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing a vector by replicating the input value. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	reverse (T_x &&x)
	Return reversed view into the specified vector or row vector. More...
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	row (T_x &&x, size_t j)
	Return the specified row of the specified kernel generator expression using start-at-1 indexing. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
int	rows (const T_x &x)
	Returns the number of rows in the specified kernel generator expression. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>
auto	rows_dot_product (T_a &&a, T_b &&b)
	Returns the dot product of rows of the specified matrices. More...
template<typename T_a , require_all_kernel_expressions_and_none_scalar_t< T_a > * = nullptr>
auto	rows_dot_self (T_a &&a)
	Returns the dot product of each row of a matrix with itself. More...
template<bool propto, typename T_y_cl , typename T_dof_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl, T_scale_cl >	scaled_inv_chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu, const T_scale_cl &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
double	sd (const T &a)
	Returns the unbiased sample standard deviation of the coefficients in the specified std vector, column vector, row vector, or matrix. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	segment (T_x &&x, size_t i, size_t n)
	Return the specified number of elements as a row/column vector starting from the specified element - 1 of the specified row/column vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	sign (const T &x)
	Returns signs of the arguments. More...
template<typename T , require_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
size_t	size (const T &m)
	Returns the size (number of the elements) of a matrix_cl or var_value<matrix_cl<T>>. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	skew_double_exponential_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	skew_double_exponential_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function. More...
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	skew_double_exponential_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	skew_double_exponential_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the log PMF of the skew double exponential distribution. More...
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	skew_normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_shape_cl &alpha)
	The log of the skew normal density for the specified scalar(s) given the specified mean(s), deviation(s) and shape(s). More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
matrix_cl< double >	softmax (const T &a)
	Return the softmax of the specified vector. More...
template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	sort_asc (T &&input)
	Sort the (row)vector in ascending order. More...
template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	sort_desc (T &&input)
	Sort the (row)vector in ascending order. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>
auto	squared_distance (const T_a &a, const T_b &b)
	Returns the squared distance. More...
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	std_normal_cdf (const T_y_cl &y)
	Returns the standard normal cumulative distribution function. More...
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	std_normal_lccdf (const T_y_cl &y)
	Returns the log standard normal complementary cumulative distribution function. More...
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	std_normal_lcdf (const T_y_cl &y)
	Returns the log standard normal complementary cumulative distribution function. More...
template<bool propto, typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	std_normal_lpdf (const T_y_cl &y)
	The log of the normal density for the specified scalar(s) given a location of 0 and a scale of 1. More...
template<bool propto, typename T_y_cl , typename T_dof_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl >	student_t_lpdf (const T_y_cl &y, const T_dof_cl &nu, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	sub_col (T_x &&x, size_t i, size_t j, size_t nrows)
	Return a 1 x ncols subrow starting at (i-1, j-1). More...
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	sub_row (T_x &&x, size_t i, size_t j, size_t ncols)
	Return a 1 x ncols subrow starting at (i-1, j-1). More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	symmetrize_from_lower_tri (T_x &&x)
	Return a symmetric matrix using elements from the upper triangular part of the input matrix. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	symmetrize_from_upper_tri (T_x &&x)
	Return a symmetric matrix using elements from the lower triangular part of the input matrix. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	tail (T_x &&x, size_t n)
	Return the specified number of elements as a vector or row vector (same as input) from the back of the specified vector or row vector. More...
template<typename T_A , typename = require_all_kernel_expressions_and_none_scalar_t<T_A>>
matrix_cl< typename std::decay_t< T_A >::Scalar >	tcrossprod (T_A &&A)
	Returns the result of post-multiplying a matrix by its own transpose. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	to_array_1d (T_x &&x)
	Returns input matrix reshaped into a vector. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
T_x	to_array_2d (T_x &&x)
	Returns input matrix converted into a nested std vector. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
T_x	to_matrix (T_x &&x)
	Returns input matrix. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
matrix_cl< return_type_t< T_x > >	to_matrix (const T_x &x, int m, int n)
	Returns a matrix representation of a vector or matrix in column-major order with the specified number of rows and columns. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	to_matrix (const T_x &x, int m, int n, bool col_major) -> decltype(to_matrix(x, m, n))
	Returns a matrix representation of the vector or matrix in column-major or row major order with the specified number of rows and columns. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	to_row_vector (T_x &&x)
	Returns input matrix reshaped into a row vector. More...
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	to_vector (T_x &&x)
	Returns input matrix reshaped into a vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
value_type_t< T >	trace (const T &m)
	Calculates trace (sum of diagonal) of given kernel generator expression. More...
template<typename T , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< U > * = nullptr>
auto	ub_constrain (T &&x, U &&ub)
	Return the upper-bounded value for the specified unconstrained matrix and upper bound. More...
template<typename T , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub, return_type_t< T, U > &lp)
	Return the upper-bounded value for the specified unconstrained matrix and upper bound. More...
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	uniform_cdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the uniform cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	uniform_lccdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the log uniform complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	uniform_lcdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the log uniform cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	uniform_lpdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	The log of a uniform density for the given y, lower, and upper bound. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	unit_vector_constrain (T_x &&x)
	Return the unit length vector corresponding to the given free vector. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
auto	unit_vector_constrain (T_x &&x, double &lp)
	Return the unit length vector corresponding to the given free vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
double	variance (const T &a)
	Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	weibull_cdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull cumulative distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	weibull_lccdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull log cumulative complementary distribution function for the given location, and scale. More...
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	weibull_lcdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	weibull_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the Weibull log probability density for the given location and scale. More...
template<bool need_Q = true>
void	qr_decomposition_cl (const matrix_cl< double > &A, matrix_cl< double > &Q, matrix_cl< double > &R, int r=100)
	Calculates QR decomposition of A using the block Householder algorithm. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	acos (const var_value< T > &A)
	Returns the elementwise acos() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	acosh (const var_value< T > &A)
	Returns the elementwise acosh() of a var_value<matrix_cl<double>> in radians. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
auto	add (T_a &&a, T_b &&b)
	Addition of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
auto	operator+ (const T_a &a, const T_b &b)
	Addition of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T_m , typename T_a , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_m > * = nullptr, require_all_prim_or_rev_kernel_expression_t< T_a > * = nullptr, require_any_var_t< T_m, T_a > * = nullptr>
auto	add_diag (const T_m &mat, const T_a &to_add)
	Returns a Matrix with values added along the main diagonal. More...
template<typename... T_results>
adjoint_results_cl< T_results... >	adjoint_results (T_results &&...results)
	Deduces types for constructing adjoint_results_cl object. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	append_col (T_a &&a, T_b &&b)
	Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	append_row (T_a &&a, T_b &&b)
	Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>
auto	as_column_vector_or_scalar (const T &m)
	Converts kernel generator expression row or column vector to a column vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	asin (const var_value< T > &A)
	Returns the elementwise asin() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	asinh (const var_value< T > &A)
	Returns the elementwise asinh() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	atan (const var_value< T > &A)
	Returns the elementwise atan() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	atanh (const var_value< T > &A)
	Returns the elementwise atanh() of a var_value<matrix_cl<double>> in radians. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
auto	beta (T_a &&a, T_b &&b)
	Return the elementwise beta() on two input kernel generator expression. More...
template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>
auto	block (const T &m, size_t i, size_t j, size_t nrows, size_t ncols)
	Return a nrows x ncols submatrix starting at (i-1, j-1). More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	cbrt (const var_value< T > &A)
	Returns the elementwise cbrt() of the input var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	ceil (const var_value< T > &A)
	Returns the elementwise ceil() of the specified variable. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	cholesky_decompose (const var_value< T > &A)
	Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric reverse mode matrix on the OpenCL device. More...
template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	columns_dot_product (T1 &&v1, T2 &&v2)
	Returns the dot product of columns of the specified matrices. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	columns_dot_self (const var_value< T > &v)
	Returns the dot product of each column of a matrix with itself. More...
template<typename T >
var_value< matrix_cl< value_type_t< T > > >	to_matrix_cl (const var_value< T > &a)
	Copies the source var containing Eigen matrices to destination var that has data stored on the OpenCL device. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
var_value< matrix_cl< value_type_t< T > > >	to_matrix_cl (const std::vector< var_value< T >> &a)
	Copies the source std::vector of vars to a destination var that has data stored on the OpenCL device. More...
template<typename T , require_eigen_vt< is_var, T > * = nullptr>
var_value< matrix_cl< value_type_t< value_type_t< T > > > >	to_matrix_cl (const T &src)
	Copies the source Eigen matrix of vars to the destination matrix that is stored on the OpenCL device. More...
template<typename T , require_eigen_vt< is_var, T > * = nullptr>
var_value< matrix_cl< value_type_t< value_type_t< T > > > >	to_matrix_cl (const std::vector< T > &src)
	Copies the source vector of Eigen matrices of vars to the destination matrix that is stored on the OpenCL device. More...
template<typename T_dst , typename T , require_var_vt< is_eigen, T_dst > * = nullptr, require_all_kernel_expressions_t< T > * = nullptr>
T_dst	from_matrix_cl (const var_value< T > &a)
	Copies the source var that has data stored on the OpenCL device to destination var containing Eigen matrix. More...
template<typename T_dst , typename T , require_std_vector_vt< is_var, T_dst > * = nullptr, require_all_stan_scalar_t< value_type_t< T_dst >> * = nullptr, require_all_kernel_expressions_t< T > * = nullptr>
T_dst	from_matrix_cl (const var_value< T > &a)
	Copies the source var that has data stored on the OpenCL device to destination std::vector containing vars. More...
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	from_matrix_cl (const var_value< T > &src)
	Copies the source var that has data stored on the OpenCL device to destination Eigen matrix containing vars. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	cos (const var_value< T > &A)
	Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	cosh (const var_value< T > &A)
	Returns the elementwise cosh() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	crossprod (const var_value< T > &M)
	Returns the result of pre-multiplying a matrix by its own transpose. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	cumulative_sum (const var_value< T > &A)
	Return the cumulative sum of the specified vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	diag_matrix (const var_value< T > &v)
	Return a square diagonal matrix with the specified vector of coefficients as the diagonal values. More...
template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	diag_post_multiply (T1 &&v1, T2 &&v2)
	Return the product of a matrix and the diagonal matrix formed from the vector or row_vector. More...
template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	diag_pre_multiply (T1 &&v1, T2 &&v2)
	Return the product of the diagonal matrix formed from the vector or row_vector and a matrix. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	diagonal (const var_value< T > &M)
	Return a column vector of the diagonal elements of the specified matrix. More...
var_value< matrix_cl< double > >	digamma (const var_value< matrix_cl< double >> &A)
	Returns the elementwise digamma() of a var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_stan_scalar_t< T_b > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	divide (T_a &&a, T_b &&b)
	Elementwise division of a kernel generator expression and a scalar. More...
template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
var	dot_product (T1 &&v1, T2 &&v2)
	Returns the dot product. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	dot_self (const var_value< T > &v)
	Returns the dot product of a vector of var with itself. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	elt_divide (T_a &&a, T_b &&b)
	Elementwise division of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	elt_multiply (T_a &&a, T_b &&b)
	Elementwise multiplication of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	erf (const var_value< T > &A)
	Returns the elementwise erf() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	erfc (const var_value< T > &A)
	Returns the elementwise erfc() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	exp (const var_value< T > &A)
	Returns the elementwise exp() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	exp2 (const var_value< T > &A)
	Returns the elementwise exp2() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	expm1 (const var_value< T > &A)
	Returns the elementwise expm1() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	fabs (const var_value< T > &A)
	Returns the elementwise fabs() of the input var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	fdim (T_a &&a, T_b &&b)
	Return the positive difference between the first variable's the value and the second's (C99, C++11). More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	floor (const var_value< T > &A)
	Returns the elementwise floor() of the input var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	fmax (T_a &&a, T_b &&b)
	Return the greater of the two specified arguments. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	fmin (T_a &&a, T_b &&b)
	Return the lesser of the two specified arguments. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	fmod (T_a &&a, T_b &&b)
	Return the floating point remainder after dividing the first variable by the second (cmath). More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	hypot (T_a &&a, T_b &&b)
	Returns the elementwise hypot() of the input. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv (const var_value< T > &A)
	Returns the elementwise inv() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv_cloglog (const var_value< T > &A)
	Returns the elementwise inv_cloglog() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv_logit (const var_value< T > &A)
	Returns the elementwise inv_logit() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv_Phi (const var_value< T > &A)
	Returns the elementwise inv_Phi() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv_sqrt (const var_value< T > &A)
	Returns the elementwise inv_sqrt() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	inv_square (const var_value< T > &A)
	Returns the elementwise inv_square() of a var_value<matrix_cl<double>>. More...
template<typename T_x , typename T_lb , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb > * = nullptr, require_any_var_t< T_x, T_lb > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb > * = nullptr>
var_value< matrix_cl< double > >	lb_constrain (T_x &&x, T_lb &&lb)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T_x , typename T_lb , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb > * = nullptr, require_any_var_t< T_x, T_lb > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb > * = nullptr>
var_value< matrix_cl< double > >	lb_constrain (T_x &&x, T_lb &&lb, var &lp)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
auto	lbeta (T_a &&a, T_b &&b)
	Return the elementwise lbeta() on two input kernel generator expression. More...
template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_st_integral< T_b > * = nullptr>
var_value< matrix_cl< double > >	ldexp (const var_value< T_a > &a, T_b &&b)
	Returns the elementwise ldexp() of the input var_value<matrix_cl<double>> and kernel generator expression. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	lgamma (const var_value< T > &A)
	Returns the elementwise lgamma() of a var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	lmultiply (T_a &&a, T_b &&b)
	Returns the elementwise lmultiply() of the input. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log (const var_value< T > &A)
	Returns the elementwise log() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log10 (const var_value< T > &A)
	Returns the elementwise log10() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log1m (const var_value< T > &A)
	Returns the elementwise log1m() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log1m_exp (const var_value< T > &A)
	Returns the elementwise log1m_exp() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log1m_inv_logit (const var_value< T > &A)
	Returns the elementwise log1m_inv_logit() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log1p (const var_value< T > &A)
	Returns the elementwise log1p() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log1p_exp (const var_value< T > &A)
	Returns the elementwise log1p_exp() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log2 (const var_value< T > &A)
	Returns the elementwise log2() of a var_value<matrix_cl<double>>. More...
template<typename T_x , typename T_y , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_any_var_t< T_x, T_y > * = nullptr, require_any_not_stan_scalar_t< T_x, T_y > * = nullptr>
var_value< matrix_cl< double > >	log_diff_exp (T_x &&x, T_y &&y)
	Returns the natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log_inv_logit (const var_value< T > &A)
	Returns the elementwise log_inv_logit() of a var_value<matrix_cl<double>>. More...
template<typename T_x , typename T_y , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_any_var_t< T_x, T_y > * = nullptr, require_any_not_stan_scalar_t< T_x, T_y > * = nullptr>
var_value< matrix_cl< double > >	log_inv_logit_diff (T_x &&x, T_y &&y)
	Returns the natural logarithm of the difference of the inverse logits of the specified arguments. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	log_softmax (const var_value< T > &A)
	Return log of the softmax of the specified vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	log_sum_exp (const var_value< T > &A)
	Return the log of the sum of the exponentiated values of the specified matrix of values. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	logit (const var_value< T > &A)
	Returns the elementwise logit() of a var_value<matrix_cl<double>>. More...
template<typename T_x , typename T_lb , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb, T_ub > * = nullptr, require_any_var_t< T_x, T_lb, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb, T_ub > * = nullptr>
var_value< matrix_cl< double > >	lub_constrain (T_x &&x, T_lb &&lb, T_ub &&ub)
	Return the bounded value for the specified unconstrained input and specified bounds. More...
template<typename T_x , typename T_lb , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb, T_ub > * = nullptr, require_any_var_t< T_x, T_lb, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb, T_ub > * = nullptr>
var_value< matrix_cl< double > >	lub_constrain (T_x &&x, T_lb &&lb, T_ub &&ub, var &lp)
	Return the bounded value for the specified unconstrained input and specified bounds. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	matrix_power (const var_value< T > &M, const int n)
	Returns the nth power of the specific matrix. More...
template<typename T1 , typename T2 , require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	mdivide_left_tri_low (T1 &&A, T2 &&b)
	Returns the solution of the system Ax=b when A is lower triangular. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	mdivide_left_tri_low (const var_value< T > &A)
	Returns the solution of the system Ax=b when A is triangular and b=I. More...
template<typename T1 , typename T2 , require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	mdivide_right_tri_low (T2 &&b, T1 &&A)
	Returns the solution of the system Ax=b when A is lower triangular. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
auto	multiply (T_a &&A, T_b &&B)
	Matrix multiplication of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
auto	operator* (const T_a &A, const T_b &B)
	Matrix multiplication of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T1 , typename T2 , require_stan_scalar_t< T1 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>
auto	multiply (const T1 &A, T2 &&B)
	Return matrix multiplied by a scalar. More...
template<typename T1 , typename T2 , require_stan_scalar_t< T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>
auto	multiply (const T1 &A, const T2 &B)
	Return matrix multiplied by a scalar. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	multiply_log (T_a &&a, T_b &&b)
	Returns the elementwise multiply_log() of the input. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	multiply_lower_tri_self_transpose (const var_value< T > &A)
	Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose. More...
template<typename T , typename M , typename S , require_all_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr, require_any_var_t< T, M, S > * = nullptr>
var_value< matrix_cl< double > >	offset_multiplier_constrain (T &&A, M &&mu, S &&sigma)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<typename T , typename M , typename S , require_all_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr, require_any_var_t< T, M, S > * = nullptr>
var_value< matrix_cl< double > >	offset_multiplier_constrain (T &&A, M &&mu, S &&sigma, var &lp)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	operator- (const var_value< T > &M)
	Returns the unary minus of the input. More...
template<typename T , require_var_vt< is_kernel_expression_and_not_scalar, T > * = nullptr>
T	operator+ (T &&M)
	Returns the unary plus of the input. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	Phi (const var_value< T > &A)
	Returns the elementwise Phi() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	Phi_approx (const var_value< T > &A)
	Returns the elementwise Phi_approx() of a var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
var_value< matrix_cl< double > >	pow (T_a &&a, T_b &&b)
	Return the first argument raised to the power of the second argument. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	prod (const var_value< T > &x)
	Returns the prod of the coefficients of the specified matrix on the OpenCL device. More...
template<typename T_ret , require_var_vt< is_matrix_cl, T_ret > * = nullptr>
var_value< matrix_cl< double > >	rep_matrix (const var &A, int n, int m)
	Creates a matrix_cl by replicating the given value of arithmetic type. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	rep_matrix (const var_value< T > &A, int m)
	Creates a matrix_cl by replicating the input vector or row_vector. More...
template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>
auto	reverse (const T &m)
	Return reversed view into the specified vector or row vector. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	round (const var_value< T > &A)
	Returns the elementwise round() of the input var_value<matrix_cl<double>>. More...
template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
var_value< matrix_cl< double > >	rows_dot_product (T1 &&v1, T2 &&v2)
	Returns the dot product of rows of the specified matrices. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	rows_dot_self (const var_value< T > &v)
	Returns the dot product of each row of a matrix with itself. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	sd (const var_value< T > &A)
	Return the sample standard deviation of the var_value matrix. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	sin (const var_value< T > &A)
	Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	sinh (const var_value< T > &A)
	Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	softmax (const var_value< T > &A)
	Returns softmax of given argument. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	sqrt (const var_value< T > &A)
	Returns the elementwise sqrt() of a var_value<matrix_cl<double>>. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	square (const var_value< T > &A)
	Returns the elementwise square() of a var_value<matrix_cl<double>>. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
var_value< double >	squared_distance (T_a &&a, T_b &&b)
	Returns the squared distance. More...
template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>
auto	subtract (T_a &&a, T_b &&b)
	Subtraction of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>
auto	operator- (const T_a &a, const T_b &b)
	Subtraction of two reverse mode matrices and/or kernel generator expressions. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	sum (const var_value< T > &x)
	Returns the sum of the coefficients of the specified matrix on the OpenCL device. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	symmetrize_from_lower_tri (const var_value< T > &A)
	Return a symmetric matrix using elements from the upper triangular part of the input matrix. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	symmetrize_from_upper_tri (const var_value< T > &A)
	Return a symmetric matrix using elements from the upper triangular part of the input matrix. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	tan (const var_value< T > &A)
	Returns the elementwise tan() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	tanh (const var_value< T > &A)
	Returns the elementwise tanh() of a var_value<matrix_cl<double>> in radians. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	tcrossprod (const var_value< T > &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	tgamma (const var_value< T > &A)
	Returns the elementwise tgamma() of a var_value<matrix_cl<double>>. More...
template<typename T , require_matrix_cl_t< T > * = nullptr>
arena_t< T >	to_arena (const T &a)
	Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered. More...
template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>
var_value< matrix_cl< double > >	to_matrix (const var_value< T_x > &x, int m, int n)
	Returns a matrix representation of a vector or matrix in column-major order with the specified number of rows and columns. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	trace (const var_value< T > &x)
	Returns the trace (sum of diagonal) of the specified matrix on the OpenCL device. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
auto	transpose (const var_value< T > &M)
	Transposes a matrix. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	trunc (const var_value< T > &A)
	Returns the elementwise trunc() of the input var_value<matrix_cl<double>>. More...
template<typename T_x , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_ub > * = nullptr, require_any_var_t< T_x, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_ub > * = nullptr>
var_value< matrix_cl< double > >	ub_constrain (T_x &&x, T_ub &&ub)
	Return the upper-bounded value for the specified unconstrained input and specified upper bound. More...
template<typename T_x , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_ub > * = nullptr, require_any_var_t< T_x, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_ub > * = nullptr>
var_value< matrix_cl< double > >	ub_constrain (T_x &&x, T_ub &&ub, var &lp)
	Return the upper-bounded value for the specified unconstrained input and specified upper bound. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	unit_vector_constrain (const var_value< T > &A)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	unit_vector_constrain (const var_value< T > &A, var &lp)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var	variance (const var_value< T > &A)
	Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero. More...
template<bool need_eigenvectors = true>
void	symmetric_eigensolver (const matrix_cl< double > &A, matrix_cl< double > &eigenvalues, matrix_cl< double > &eigenvectors)
template<typename T >
ref_type_for_opencl_t< T && >	to_ref_for_opencl (T &&a)
	Converts given Eigen expression into one that can be directly copied to an OpenCL device to create matrix_cl. More...
template<matrix_cl_view matrix_view = matrix_cl_view::Entire, typename T , require_matrix_cl_st< std::is_floating_point, T > * = nullptr>
plain_type_t< T >	tri_inverse (const T &A)
	Computes the inverse of a triangular matrix. More...
tbb::task_scheduler_init &	init_threadpool_tbb (int n_threads=0)
	Initialize the Intel TBB threadpool and global scheduler through the tbb::task_scheduler_init object. More...
template<typename U , typename V >
complex_return_t< U, V >	operator+ (const std::complex< U > &x, const std::complex< V > &y)
	Return the sum of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator+ (const std::complex< U > &x, const V &y)
	Return the sum of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator+ (const U &x, const std::complex< V > &y)
	Return the sum of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator/ (const std::complex< U > &x, const std::complex< V > &y)
	Return the quotient of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator/ (const std::complex< U > &x, const V &y)
	Return the quotient of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator/ (const U &x, const std::complex< V > &y)
	Return the quotient of the arguments. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator== (const std::complex< U > &x, const std::complex< V > &y)
	Return true if the complex numbers have equal imaginary and complex parts. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator== (const std::complex< U > &x, const V &y)
	Return true if the first argument's real part is equal to the second argument and the first argument's imaginary part is zero. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator== (const U &x, const std::complex< V > &y)
	Return true if the first argument is equal to the real part of the second argument and the imaginary part of the second argument is zero. More...
template<typename U , require_autodiff_t< U > >
std::complex< U >	operator- (const std::complex< U > &x)
	Return the negation of the argument. More...
template<typename U , typename V >
complex_return_t< U, V >	operator* (const std::complex< U > &x, const std::complex< V > &y)
	Return the product of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator* (const std::complex< U > &x, const V &y)
	Return the product of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator* (const U &x, const std::complex< V > &y)
	Return the product of the arguments. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator!= (const std::complex< U > &x, const std::complex< V > &y)
	Return true if the complex numbers have unequal imaginary or complex parts. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator!= (const std::complex< U > &x, const V &y)
	Return true if the first argument's real part is unequal to the second argument or the first argument's imaginary part is unequal to zero. More...
template<typename U , typename V , typename = require_any_autodiff_t<U, V>>
bool	operator!= (const U &x, const std::complex< V > &y)
	Return true if the first argument is unequal to the real part of the second argument or the imaginary part of the second argument is nonzero. More...
template<typename U , require_autodiff_t< U > >
std::complex< U >	operator+ (const std::complex< U > &x)
	Return the argument. More...
template<typename U , typename V >
complex_return_t< U, V >	operator- (const std::complex< U > &x, const std::complex< V > &y)
	Return the difference of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator- (const std::complex< U > &x, const V &y)
	Return the difference of the arguments. More...
template<typename U , typename V >
complex_return_t< U, V >	operator- (const U &x, const std::complex< V > &y)
	Return the difference of the arguments. More...
template<typename T_a1 , typename T_a2 , typename T_b1 , typename T_z >
void	check_2F1_converges (const char *function, const T_a1 &a1, const T_a2 &a2, const T_b1 &b1, const T_z &z)
	Check if the hypergeometric function (2F1) called with supplied arguments will converge, assuming arguments are finite values. More...
template<typename T_a1 , typename T_a2 , typename T_a3 , typename T_b1 , typename T_b2 , typename T_z >
void	check_3F2_converges (const char *function, const T_a1 &a1, const T_a2 &a2, const T_a3 &a3, const T_b1 &b1, const T_b2 &b2, const T_z &z)
	Check if the hypergeometric function (3F2) called with supplied arguments will converge, assuming arguments are finite values. More...
template<typename T_y , typename T_low , typename T_high >
void	check_bounded (const char *function, const char *name, const T_y &y, const T_low &low, const T_high &high)
	Check if the value is between the low and high values, inclusively. More...
template<typename Mat , require_matrix_t< Mat > * = nullptr>
void	check_cholesky_factor (const char *function, const char *name, const Mat &y)
	Throw an exception if the specified matrix is not a valid Cholesky factor. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_cholesky_factor (const char *function, const char *name, const StdVec &y)
	Throw an exception if the specified matrix is not a valid Cholesky factor. More...
template<typename Mat , require_matrix_t< Mat > * = nullptr>
void	check_cholesky_factor_corr (const char *function, const char *name, const Mat &y)
	Throw an exception if the specified matrix is not a valid Cholesky factor of a correlation matrix. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_cholesky_factor_corr (const char *function, const char *name, const StdVec &y)
	Throw an exception if the specified matrix is not a valid Cholesky factor of a correlation matrix. More...
template<typename T_y , require_any_t< is_matrix< T_y >, is_prim_or_rev_kernel_expression< T_y >> * = nullptr>
void	check_column_index (const char *function, const char *name, const T_y &y, size_t i)
	Check if the specified index is a valid column of the matrix. More...
template<typename T >
void	check_consistent_size (const char *function, const char *name, const T &x, size_t expected_size)
	Check if x is consistent with size expected_size. More...
void	check_consistent_sizes (const char *)
	Trivial no input case, this function is a no-op. More...
template<typename T1 >
void	check_consistent_sizes (const char *, const char *, const T1 &)
	Base case of recursion, this function is a no-op. More...
template<typename T1 , typename T2 , typename... Ts>
void	check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const Ts &...names_and_xs)
	Check that the inputs are of consistent size. More...
void	check_consistent_sizes_mvt (const char *)
	Trivial no input case, this function is a no-op. More...
template<typename T1 >
void	check_consistent_sizes_mvt (const char *, const char *, const T1 &)
	Base case of recursion, this function is a no-op. More...
template<typename T1 , typename T2 , typename... Ts>
void	check_consistent_sizes_mvt (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const Ts &...names_and_xs)
	Check that the provided multivariate inputs are of consistent size with each other. More...
template<typename Mat , require_matrix_t< Mat > * = nullptr>
void	check_corr_matrix (const char *function, const char *name, const Mat &y)
	Throw an exception if the specified matrix is not a valid correlation matrix. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_corr_matrix (const char *function, const char *name, const StdVec &y)
	Throw an exception if the specified matrix is not a valid correlation matrix. More...
template<typename Mat , require_matrix_t< Mat > * = nullptr>
void	check_cov_matrix (const char *function, const char *name, const Mat &y)
	Throw an exception if the specified matrix is not a valid covariance matrix. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_cov_matrix (const char *function, const char *name, const StdVec &y)
	Throw an exception if the specified matrix is not a valid covariance matrix. More...
template<typename T_y >
void	check_finite (const char *function, const char *name, const T_y &y)
	Return true if all values in y are finite. More...
void	check_flag_sundials (int flag, const char *func_name)
	Throws a std::runtime_error exception when a Sundial function fails (i.e. More...
void	check_flag_kinsol (int flag, long int max_num_steps)
	Throws an exception message when the function KINSol() (call to the solver) fails. More...
template<typename T_y , typename T_low , require_all_stan_scalar_t< T_y, T_low > * = nullptr, typename... Idxs>
void	check_greater (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not strictly greater than low. More...
template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_low > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_low > * = nullptr, typename... Idxs>
void	check_greater (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not strictly greater than each element of low. More...
template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_low > * = nullptr, typename... Idxs>
void	check_greater (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not strictly greater than each element of low. More...
template<typename T_y , typename T_low , require_all_stan_scalar_t< T_y, T_low > * = nullptr, typename... Idxs>
void	check_greater_or_equal (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not greater or equal than low. More...
template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_low > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_low > * = nullptr, typename... Idxs>
void	check_greater_or_equal (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not greater or equal than each element of low. More...
template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_low > * = nullptr, typename... Idxs>
void	check_greater_or_equal (const char *function, const char *name, const T_y &y, const T_low &low, Idxs...idxs)
	Throw an exception if y is not greater or equal than each element of low. More...
template<typename T >
void	check_ldlt_factor (const char *function, const char *name, LDLT_factor< T > &A)
	Raise domain error if the specified LDLT factor is invalid. More...
template<typename T_y , typename T_high , require_all_stan_scalar_t< T_y, T_high > * = nullptr, typename... Idxs>
void	check_less (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not strictly less than high. More...
template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_high > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_high > * = nullptr, typename... Idxs>
void	check_less (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not strictly less than each element of high. More...
template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_high > * = nullptr, typename... Idxs>
void	check_less (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not strictly less than each element of high. More...
template<typename T_y , typename T_high , require_all_stan_scalar_t< T_y, T_high > * = nullptr, typename... Idxs>
void	check_less_or_equal (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not less than high. More...
template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_high > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_high > * = nullptr, typename... Idxs>
void	check_less_or_equal (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not less than each element of high. More...
template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_high > * = nullptr, typename... Idxs>
void	check_less_or_equal (const char *function, const char *name, const T_y &y, const T_high &high, Idxs...idxs)
	Throw an exception if y is not less than each element of high. More...
template<typename T_y , require_eigen_t< T_y > * = nullptr>
void	check_lower_triangular (const char *function, const char *name, const T_y &y)
	Check if the specified matrix is lower triangular. More...
template<typename T1 , typename T2 , require_all_not_matrix_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
void	check_matching_dims (const char *function, const char *name1, const T1 &y1, const char *name2, const T2 &y2)
	Check if the two containers have the same dimensions. More...
template<bool check_compile, typename Mat1 , typename Mat2 , typename = require_all_eigen_t<Mat1, Mat2>>
void	check_matching_dims (const char *function, const char *name1, const Mat1 &y1, const char *name2, const Mat2 &y2)
	Check if the two matrices are of the same size. More...
template<typename T_y1 , typename T_y2 >
void	check_matching_sizes (const char *function, const char *name1, const T_y1 &y1, const char *name2, const T_y2 &y2)
	Check if two structures at the same size. More...
template<typename T1 , typename T2 >
void	check_multiplicable (const char *function, const char *name1, const T1 &y1, const char *name2, const T2 &y2)
	Check if the matrices can be multiplied. More...
template<typename T_y >
void	check_nonnegative (const char *function, const char *name, const T_y &y)
	Check if y is non-negative. More...
template<typename T_y >
void	check_nonzero_size (const char *function, const char *name, const T_y &y)
	Check if the specified matrix/vector is of non-zero size. More...
template<typename T_y >
void	check_not_nan (const char *function, const char *name, const T_y &y)
	Check if y is not NaN. More...
template<typename T_y , require_vector_t< T_y > * = nullptr, require_not_std_vector_t< T_y > * = nullptr>
void	check_ordered (const char *function, const char *name, const T_y &y)
	Throw an exception if the specified vector is not sorted into strictly increasing order. More...
template<typename T_y , require_std_vector_vt< is_stan_scalar, T_y > * = nullptr>
void	check_ordered (const char *function, const char *name, const T_y &y)
	Throw an exception if the specified vector is not sorted into strictly increasing order. More...
template<typename EigMat , require_matrix_t< EigMat > * = nullptr>
void	check_pos_definite (const char *function, const char *name, const EigMat &y)
	Check if the specified square, symmetric matrix is positive definite. More...
template<typename Derived >
void	check_pos_definite (const char *function, const char *name, const Eigen::LDLT< Derived > &cholesky)
	Check if the specified LDLT decomposition of a matrix is positive definite. More...
template<typename Derived >
void	check_pos_definite (const char *function, const char *name, const Eigen::LLT< Derived > &cholesky)
	Check if the specified LLT decomposition was successful. More...
template<typename EigMat , require_matrix_t< EigMat > * = nullptr>
void	check_pos_semidefinite (const char *function, const char *name, const EigMat &y)
	Check if the specified matrix is positive definite. More...
template<typename Derived >
void	check_pos_semidefinite (const char *function, const char *name, const Eigen::LDLT< Derived > &cholesky)
	Check if the specified matrix is positive semidefinite. More...
template<typename T_y >
void	check_positive (const char *function, const char *name, const T_y &y)
	Check if y is positive. More...
void	check_positive (const char *function, const char *name, const char *expr, int size)
	Check if size is positive. More...
template<typename T_y >
void	check_positive_finite (const char *function, const char *name, const T_y &y)
	Check if y is positive and finite. More...
template<typename Vec , require_vector_t< Vec > * = nullptr, require_not_std_vector_t< Vec > * = nullptr>
void	check_positive_ordered (const char *function, const char *name, const Vec &y)
	Throw an exception if the specified the vector contains negative values or is not sorted into strictly increasing order. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_positive_ordered (const char *function, const char *name, const StdVec &y)
	Throw an exception if any of the vectors in a standard vector contains negative values or is not sorted into strictly increasing order. More...
void	check_range (const char *function, const char *name, int max, int index, int nested_level, const char *error_msg)
	Check if specified index is within range. More...
void	check_range (const char *function, const char *name, int max, int index, const char *error_msg)
	Check if specified index is within range. More...
void	check_range (const char *function, const char *name, int max, int index)
	Check if specified index is within range. More...
template<typename T_y , typename = require_matrix_t<T_y>>
void	check_row_index (const char *function, const char *name, const T_y &y, size_t i)
	Check if the specified index is a valid row of the matrix This check is 1-indexed by default. More...
template<typename T , require_matrix_t< T > * = nullptr>
void	check_simplex (const char *function, const char *name, const T &theta)
	Throw an exception if the specified vector is not a simplex. More...
template<typename T_size1 , typename T_size2 >
void	check_size_match (const char *function, const char *name_i, T_size1 i, const char *name_j, T_size2 j)
	Check if the provided sizes match. More...
template<typename T_size1 , typename T_size2 >
void	check_size_match (const char *function, const char *expr_i, const char *name_i, T_size1 i, const char *expr_j, const char *name_j, T_size2 j)
	Check if the provided sizes match. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
void	check_sorted (const char *function, const char *name, const EigVec &y)
	Check if the specified vector is sorted into increasing order (repeated values are okay). More...
template<typename T_y >
void	check_sorted (const char *function, const char *name, const std::vector< T_y > &y)
	Check if the specified vector is sorted into increasing order (repeated values are okay). More...
template<typename T_y , require_any_t< is_matrix< T_y >, is_prim_or_rev_kernel_expression< T_y >> * = nullptr>
void	check_square (const char *function, const char *name, const T_y &y)
	Check if the specified matrix is square. More...
template<typename T >
void	check_std_vector_index (const char *function, const char *name, const std::vector< T > &y, int i)
	Check if the specified index is valid in std vector This check is 1-indexed by default. More...
template<typename EigMat , require_matrix_t< EigMat > * = nullptr>
void	check_symmetric (const char *function, const char *name, const EigMat &y)
	Check if the specified matrix is symmetric. More...
template<typename Vec , require_vector_t< Vec > * = nullptr, require_not_std_vector_t< Vec > * = nullptr>
void	check_unit_vector (const char *function, const char *name, const Vec &theta)
	Throw an exception if the specified vector does not have unit Euclidiean length. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>
void	check_unit_vector (const char *function, const char *name, const StdVec &theta)
	Throw an exception if the each element in a standard vector does not have unit Euclidiean length. More...
template<typename Mat , require_any_t< is_matrix< Mat >, is_prim_or_rev_kernel_expression< Mat >> * = nullptr>
void	check_vector (const char *function, const char *name, const Mat &x)
	Check the input is either a row vector or column vector or a matrix with a single row or column. More...
template<typename T , require_any_t< is_vector< T >, is_prim_or_rev_kernel_expression< T >> * = nullptr>
void	check_vector_index (const char *function, const char *name, const T &y, size_t i)
	Check if the specified index is a valid element of the row or column vector This check is 1-indexed by default. More...
template<typename T >
void	domain_error (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
template<typename T >
void	domain_error (const char *function, const char *name, const T &y, const char *msg1)
template<typename T >
void	domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
template<typename T >
void	domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1)
template<typename F , typename T , typename... Indexings, require_stan_scalar_t< T > * = nullptr>
void	elementwise_check (const F &is_good, const char *function, const char *name, const T &x, const char *must_be, const Indexings &...indexings)
	Check that the predicate holds for the value of x. More...
template<typename F , typename T , typename... Indexings, require_eigen_t< T > * = nullptr, std::enable_if_t<(Eigen::internal::traits< T >::Flags &Eigen::LinearAccessBit)||T::IsVectorAtCompileTime > * = nullptr>
void	elementwise_check (const F &is_good, const char *function, const char *name, const T &x, const char *must_be, const Indexings &...indexings)
	Check that the predicate holds for all elements of the value of x. More...
template<typename F , typename T >
bool	elementwise_is (const F &is_good, const T &x)
	Check that the predicate holds for the value of x, working elementwise on containers. More...
template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>
void	hmm_check (const T_omega &log_omegas, const T_Gamma &Gamma, const T_rho &rho, const char *function)
	Check arguments for hidden Markov model functions with a discrete latent state (lpdf, rng for latent states, and marginal probabilities for latent sates). More...
template<typename T >
void	invalid_argument (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
	Throw an invalid_argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument (const char *function, const char *name, const T &y, const char *msg1)
	Throw an invalid_argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
	Throw an invalid argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument_vec (const char *function, const char *name, const T &y, size_t i, const char *msg)
	Throw an invalid argument exception with a consistently formatted message. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_cholesky_factor (const EigMat &y)
	Return true if y is a valid Cholesky factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_cholesky_factor_corr (const EigMat &y)
	Return true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no zero columns, and no element in matrix is NaN. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
bool	is_column_index (const EigMat &y, size_t i)
	Return true if column index is in bounds. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_corr_matrix (const EigMat &y)
	Return true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive). More...
template<typename T >
bool	is_ldlt_factor (LDLT_factor< T > &A)
	Return true if the specified LDLT factor is invalid. More...
template<typename T_y , typename T_high >
bool	is_less_or_equal (const T_y &y, const T_high &high)
	Return true if y is less or equal to high. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_lower_triangular (const EigMat &y)
	Return true is matrix is lower triangular. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
bool	is_mat_finite (const EigMat &y)
	Return true is the specified matrix is finite. More...
template<typename EigMat1 , typename EigMat2 , require_all_matrix_t< EigMat1, EigMat2 > * = nullptr>
bool	is_matching_dims (const EigMat1 &y1, const EigMat2 &y2)
	Return true if the two matrices are of the same size. More...
template<bool check_compile, typename EigMat1 , typename EigMat2 , require_all_matrix_t< EigMat1, EigMat2 > * = nullptr>
bool	is_matching_dims (const EigMat1 &y1, const EigMat2 &y2)
	Return true if the two matrices are of the same size. More...
template<typename T_y1 , typename T_y2 >
bool	is_matching_size (const T_y1 &y1, const T_y2 &y2)
	Return true if two structures are the same size. More...
template<typename T_y >
bool	is_nonzero_size (const T_y &y)
	Returns true if the specified matrix/vector is size nonzero. More...
template<typename T_y >
bool	is_not_nan (const T_y &y)
	Return true if y is not NaN. More...
template<typename T_y >
bool	is_ordered (const std::vector< T_y > &y)
	Return true if the vector is sorted into strictly increasing order. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_pos_definite (const EigMat &y)
	Return true if the matrix is square or if the matrix has non-zero size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN. More...
template<typename Derived >
bool	is_pos_definite (const Eigen::LDLT< Derived > &cholesky)
	Return true if the matrix is positive definite. More...
template<typename Derived >
bool	is_pos_definite (const Eigen::LLT< Derived > &cholesky)
	Return true if diagonal of the L matrix is positive. More...
template<typename T_y >
bool	is_positive (const T_y &y)
	Return true if y is positive. More...
bool	is_positive (int size)
	Return true if size is positive. More...
template<typename T_y >
bool	is_scal_finite (const T_y &y)
	Return true if y is finite. More...
template<typename T_size1 , typename T_size2 >
bool	is_size_match (T_size1 i, T_size2 j)
	Return true if the provided sizes match. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
bool	is_square (const EigMat &y)
	Return true if the matrix is square. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>
bool	is_symmetric (const EigMat &y)
	Return true if the matrix is square, and no element not on the main diagonal is NaN. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
bool	is_unit_vector (const EigVec &theta)
	Return true if the vector is not a unit vector or if any element is NaN. More...
void	out_of_range (const char *function, int max, int index, const char *msg1="", const char *msg2="")
	Throw an out_of_range exception with a consistently formatted message. More...
void	system_error (const char *function, const char *name, const int &y, const char *msg1, const char *msg2)
	Throw a system error with a consistently formatted message. More...
void	system_error (const char *function, const char *name, const int &y, const char *msg1)
	Throw a system error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
	Throw a domain error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error (const char *function, const char *name, const T &y, const char *msg1)
	Throw a domain error with a consistently formatted message. More...
template<typename T , require_eigen_t< T > * = nullptr>
void	throw_domain_error_mat (const char *function, const char *name, const T &y, size_t i, size_t j, const char *msg1, const char *msg2)
	Throw a domain error with a consistently formatted message for matrices. More...
template<typename T >
void	throw_domain_error_mat (const char *function, const char *name, const T &y, size_t i, size_t j, const char *msg)
	Throw a domain error with a consistently formatted message for matrices. More...
template<typename T >
void	throw_domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
	Throw a domain error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg)
	Throw a domain error with a consistently formatted message. More...
void	validate_non_negative_index (const char *var_name, const char *expr, int val)
void	validate_positive_index (const char *var_name, const char *expr, int val)
	Check that simplex is at least size 1. More...
void	validate_unit_vector_index (const char *var_name, const char *expr, int val)
	Check that unit vector is at least size 2. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
T	abs (T x)
	Return the absolute value of the specified arithmetic argument. More...
template<typename T >
std::vector< T >	abs (const std::vector< T > &x)
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	abs (const Eigen::Matrix< T, R, C > &x)
	Return the elementwise absolute value of the specified matrix, vector, or row vector. More...
template<typename T , require_complex_t< T > * = nullptr>
auto	abs (T x)
	Return the absolute value (also known as the norm, modulus, or magnitude) of the specified complex argument. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	acos (const Container &x)
	Returns the elementwise acos() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	acos (const Container &x)
	Version of acos() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
double	acosh (double x)
	Return the inverse hyperbolic cosine of the specified value. More...
double	acosh (int x)
	Integer version of acosh. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	acosh (const T &x)
	Return the elementwise application of acosh() to specified argument container. More...
template<typename ScalarA , typename ScalarB , require_all_stan_scalar_t< ScalarA, ScalarB > * = nullptr, require_all_not_var_t< ScalarA, ScalarB > * = nullptr>
return_type_t< ScalarA, ScalarB >	add (const ScalarA &a, const ScalarB &b)
	Return the sum of the specified scalars. More...
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>
auto	add (const Mat1 &m1, const Mat2 &m2)
	Return the sum of the specified matrices. More...
template<typename Mat , typename Scal , require_eigen_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr, require_all_not_st_var< Mat, Scal > * = nullptr>
auto	add (const Mat &m, const Scal c)
	Return the sum of the specified matrix and specified scalar. More...
template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>
auto	add (const Scal c, const Mat &m)
	Return the sum of the specified scalar and specified matrix. More...
template<typename T_m , typename T_a , typename = require_eigen_t<T_m>, typename = require_any_t<is_eigen_vector<T_a>, is_stan_scalar<T_a>>>
Eigen::Matrix< return_type_t< T_m, T_a >, Eigen::Dynamic, Eigen::Dynamic >	add_diag (const T_m &mat, const T_a &to_add)
	Returns a Matrix with values added along the main diagonal. More...
template<typename T1 , typename T2 >
append_return_type< std::vector< T1 >, std::vector< T2 > >::type	append_array (const std::vector< T1 > &x, const std::vector< T2 > &y)
	Return the concatenation of two specified vectors in the order of the arguments. More...
template<typename T1 >
std::vector< T1 >	append_array (const std::vector< T1 > &x, const std::vector< T1 > &y)
	Return the concatenation of two specified vectors in the order of the arguments. More...
template<typename T1 , typename T2 , typename = require_all_eigen_t<T1, T2>>
auto	append_col (const T1 &A, const T2 &B)
	Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
template<typename Scal , typename RowVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_row_vector< RowVec >> * = nullptr>
Eigen::Matrix< return_type_t< Scal, RowVec >, 1, Eigen::Dynamic >	append_col (const Scal &A, const RowVec &B)
	Return the result of stacking an scalar on top of the a row vector, with the result being a row vector. More...
template<typename RowVec , typename Scal , require_t< is_eigen_row_vector< RowVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>
Eigen::Matrix< return_type_t< RowVec, Scal >, 1, Eigen::Dynamic >	append_col (const RowVec &A, const Scal &B)
	Return the result of stacking a row vector on top of the an scalar, with the result being a row vector. More...
template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr>
auto	append_row (const T1 &A, const T2 &B)
	Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
template<typename Scal , typename ColVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_col_vector< ColVec >> * = nullptr>
Eigen::Matrix< return_type_t< Scal, ColVec >, Eigen::Dynamic, 1 >	append_row (const Scal &A, const ColVec &B)
	Return the result of stacking an scalar on top of the a vector, with the result being a vector. More...
template<typename ColVec , typename Scal , require_t< is_eigen_col_vector< ColVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>
Eigen::Matrix< return_type_t< ColVec, Scal >, Eigen::Dynamic, 1 >	append_row (const ColVec &A, const Scal &B)
	Return the result of stacking a vector on top of the an scalar, with the result being a vector. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
T	as_array_or_scalar (T &&v)
	Returns specified input value. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
T &	as_array_or_scalar (T &v)
	Returns a reference to rvalue specified input value. More...
template<typename T , typename = require_eigen_t<T>, require_not_eigen_array_t< T > * = nullptr>
auto	as_array_or_scalar (T &&v)
	Converts a matrix type to an array. More...
template<typename T >
bool	as_bool (const T &x)
	Return true if the argument is not equal to zero (in the != operator sense) and false otherwise. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
T	as_column_vector_or_scalar (const T &a)
	no-op that passes the scalar More...
template<typename T , typename S >
internal::empty_broadcast_array< T, S, void > &	as_column_vector_or_scalar (internal::empty_broadcast_array< T, S, void > &a)
	No-op used when working with operands and partials. More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
T &&	as_column_vector_or_scalar (T &&a)
	no-op that returns a column vector. More...
template<typename T , require_eigen_row_vector_t< T > * = nullptr, require_not_eigen_col_vector_t< T > * = nullptr>
auto	as_column_vector_or_scalar (T &&a)
	Converts a row vector to an eigen column vector. More...
template<typename T >
auto	as_value_array_or_scalar (T &&v)
	Extract the value from an object. More...
template<typename T >
auto	as_value_column_array_or_scalar (T &&a)
	Extract the value from an object and for eigen vectors and std::vectors convert to an eigen column array and for scalars return a scalar. More...
template<typename T >
auto	as_value_column_vector_or_scalar (T &&a)
	Extract values from input argument and transform to a column vector. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	asin (const Container &x)
	Returns the elementwise asin() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	asin (const Container &x)
	Version of asin() that accepts std::vectors, Eigen Matrix/Array objects, or expressions, and containers of these. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	asinh (const T &x)
	Returns the elementwise asinh() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<int N>
void	print_mat_size (std::ostream &o)
	Helper function to return the matrix size as either "dynamic" or "1". More...
template<typename T_lhs , typename T_rhs , require_all_stan_scalar_t< T_lhs, T_rhs > * = nullptr>
void	assign (T_lhs &x, const T_rhs &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T_rhs , require_all_eigen_t< T_lhs, T_rhs > * = nullptr>
void	assign (T_lhs &&x, const T_rhs &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T_rhs >
void	assign (std::vector< T_lhs > &x, const std::vector< T_rhs > &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	atan (const Container &x)
	Returns the elementwise atan() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	atan (const Container &x)
	Version of atan() that accepts std::vectors, Eigen Matrix/Array objects, or expressions, and containers of these. More...
double	atanh (double x)
	Return the inverse hyperbolic tangent of the specified value. More...
double	atanh (int x)
	Integer version of atanh. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	atanh (const T &x)
	Return the elementwise application of atanh() to specified argument container. More...
template<typename T >
void	autocorrelation (const std::vector< T > &y, std::vector< T > &ac, Eigen::FFT< T > &fft)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocorrelation (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &ac, Eigen::FFT< T > &fft)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T >
void	autocorrelation (const std::vector< T > &y, std::vector< T > &ac)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocorrelation (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &ac)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result. More...
template<typename T >
void	autocovariance (const std::vector< T > &y, std::vector< T > &acov, Eigen::FFT< T > &fft)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocovariance (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &acov, Eigen::FFT< T > &fft)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T >
void	autocovariance (const std::vector< T > &y, std::vector< T > &acov)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocovariance (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &acov)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result. More...
template<typename T2 , require_arithmetic_t< T2 > * = nullptr>
T2	bessel_first_kind (int v, const T2 z)
	\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_not_var_matrix_t< T2 > * = nullptr>
auto	bessel_first_kind (const T1 &a, const T2 &b)
	Enables the vectorised application of the bessel first kind function, when the first and/or second arguments are containers. More...
template<typename T2 , require_arithmetic_t< T2 > * = nullptr>
T2	bessel_second_kind (int v, const T2 z)
	\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	bessel_second_kind (const T1 &a, const T2 &b)
	Enables the vectorised application of the bessel second kind function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	beta (const T1 a, const T2 b)
	Return the beta function applied to the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>
auto	beta (const T1 &a, const T2 &b)
	Enables the vectorised application of the beta function, when the first and/or second arguments are containers. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
T	binary_log_loss (int y, const T &y_hat)
	Returns the log loss function for binary classification with specified reference and response values. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_not_var_matrix_t< T2 > * = nullptr>
auto	binary_log_loss (const T1 &a, const T2 &b)
	Enables the vectorised application of the binary log loss function, when the first and/or second arguments are containers. More...
template<typename T_n , typename T_k , require_all_stan_scalar_t< T_n, T_k > * = nullptr>
return_type_t< T_n, T_k >	binomial_coefficient_log (const T_n n, const T_k k)
	Return the log of the binomial coefficient for the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	binomial_coefficient_log (const T1 &a, const T2 &b)
	Enables the vectorised application of the binomial coefficient log function, when the first and/or second arguments are containers. More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	block (const T &m, size_t i, size_t j, size_t nrows, size_t ncols)
	Return a nrows x ncols submatrix starting at (i-1, j-1). More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	cbrt (const T &x)
	Returns the elementwise cbrt() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	ceil (const Container &x)
	Returns the elementwise ceil() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T , require_eigen_t< T > * = nullptr>
plain_type_t< T >	chol2inv (const T &L)
	Returns the inverse of the matrix whose Cholesky factor is L. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>
Eigen::Matrix< value_type_t< EigVec >, Eigen::Dynamic, Eigen::Dynamic >	cholesky_corr_constrain (const EigVec &y, int K)
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
Eigen::Matrix< value_type_t< EigVec >, Eigen::Dynamic, Eigen::Dynamic >	cholesky_corr_constrain (const EigVec &y, int K, return_type_t< EigVec > &lp)
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	cholesky_corr_constrain (const T &y, int K, return_type_t< T > &lp)
	Return The cholesky of a KxK correlation matrix. More...
template<typename T , require_eigen_t< T > * = nullptr>
auto	cholesky_corr_free (const T &x)
	Overload of cholesky_corr_free() to untransform each matrix in a standard vector. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vt< is_var, EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime >	cholesky_decompose (const EigMat &m)
	Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix. More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cholesky_factor_constrain (const T &x, int M, int N)
	Return the Cholesky factor of the specified size read from the specified vector. More...
template<typename T , require_eigen_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cholesky_factor_constrain (const T &x, int M, int N, return_type_t< T > &lp)
	Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log absolute Jacobian determinant adjustment of the transform. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	cholesky_factor_constrain (const T &x, int M, int N, return_type_t< T > &lp)
	Return the Cholesky factor of the specified size read from the specified vector. More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, 1 >	cholesky_factor_free (const T &y)
	Return the unconstrained vector of parameters corresponding to the specified Cholesky factor. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	cholesky_factor_free (const T &x)
	Overload of cholesky_factor_free() to untransform each matrix in a standard vector. More...
int	choose (int n, int k)
	Return the binomial coefficient for the specified integer arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	choose (const T1 &a, const T2 &b)
	Enables the vectorised application of the binomial coefficient function, when the first and/or second arguments are containers. More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	col (const T &m, size_t j)
	Return the specified column of the specified matrix using start-at-1 indexing. More...
template<typename T , require_matrix_t< T > * = nullptr>
Eigen::Index	cols (const T &m)
	Return the number of columns in the specified matrix, vector, or row vector. More...
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>
Eigen::Matrix< return_type_t< Mat1, Mat2 >, 1, Mat1::ColsAtCompileTime >	columns_dot_product (const Mat1 &v1, const Mat2 &v2)
	Returns the dot product of columns of the specified matrices. More...
template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_var, T > * = nullptr>
Eigen::Matrix< value_type_t< T >, 1, T::ColsAtCompileTime >	columns_dot_self (const T &x)
	Returns the dot product of each column of a matrix with itself. More...
static constexpr double	e ()
	Return the base of the natural logarithm. More...
static constexpr double	egamma ()
	Return the Euler's gamma constant. More...
static constexpr double	pi ()
	Return the value of pi. More...
static constexpr double	positive_infinity ()
	Return positive infinity. More...
static constexpr double	negative_infinity ()
	Return negative infinity. More...
static constexpr double	not_a_number ()
	Return (quiet) not-a-number. More...
static constexpr double	machine_precision ()
	Returns the difference between 1.0 and the next value representable. More...
static constexpr double	log10 ()
	Returns the natural logarithm of ten. More...
static constexpr double	sqrt2 ()
	Returns the square root of two. More...
template<typename T , typename U >
T	copysign (const T &x, const U &y)
	Return the negation of the first argument if the first and second argument have different signs, otherwise return a copy of the first argument. More...
template<typename T , typename U >
T	copysign_non_zero (const T &x, const U &y)
	Return the negation of the first argument if the first and second arguments have different signs and the first argument is not zero, otherwise return a copy of the first argument. More...
template<typename T , typename U >
std::complex< T >	copysign (const std::complex< T > &x, const std::complex< U > &y)
	Return the complex number composed of the real and complex parts with signs copied from the real and complex parts of the first arguments to the real and complex parts of the second. More...
template<typename T >
plain_type_t< T >	corr_constrain (const T &x)
	Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive). More...
template<typename T_x , typename T_lp >
auto	corr_constrain (const T_x &x, T_lp &lp)
	Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive). More...
template<bool Jacobian, typename T_x , typename T_lp >
auto	corr_constrain (const T_x &x, T_lp &lp)
	Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive). More...
template<typename T >
T	corr_free (const T &y)
	Return the unconstrained scalar that when transformed to a valid correlation produces the specified value. More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	corr_matrix_constrain (const T &x, Eigen::Index k)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	corr_matrix_constrain (const T &x, Eigen::Index k, return_type_t< T > &lp)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	corr_matrix_constrain (const T &x, Eigen::Index k, return_type_t< T > &lp)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>
auto	corr_matrix_constrain (const T &y, int K, return_type_t< T > &lp)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, 1 >	corr_matrix_free (const T &y)
	Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	corr_matrix_free (const T &x)
	Overload of corr_matrix_free() to untransform each matrix in a standard vector. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	cos (const Container &x)
	Returns the elementwise cos() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	cos (const Container &x)
	Version of cos() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	cosh (const Container &x)
	Returns the elementwise cosh() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	cosh (const Container &x)
	Version of cosh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &length_scale)
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x > &x, const T_sigma &sigma, const std::vector< T_l > &length_scale)
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &length_scale)
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const std::vector< T_l > &length_scale)
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain (const T &x, Eigen::Index K)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain (const T &x, Eigen::Index K, return_type_t< T > &lp)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	cov_matrix_constrain (const T &x, Eigen::Index K, return_type_t< T > &lp)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T , require_eigen_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain_lkj (const T &x, size_t k)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values. More...
template<typename T , require_eigen_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain_lkj (const T &x, size_t k, return_type_t< T > &lp)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	cov_matrix_constrain_lkj (const T &x, size_t k, return_type_t< T > &lp)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values. More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, 1 >	cov_matrix_free (const T &y)
	The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2). More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	cov_matrix_free (const T &x)
	Overload of cov_matrix_free() to untransform each matrix in a standard vector. More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, 1 >	cov_matrix_free_lkj (const T &y)
	Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	cov_matrix_free_lkj (const T &x)
	Overload of cov_matrix_free_lkj() to untransform each matrix in a standard vector. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
auto	crossprod (const EigMat &M)
	Returns the result of pre-multiplying a matrix by its own transpose. More...
template<typename T >
const std::vector< int >	csr_extract_u (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
	Extract the NZE index for each entry from a sparse matrix. More...
template<typename T , require_eigen_dense_base_t< T > * = nullptr>
const std::vector< int >	csr_extract_u (const T &A)
	Extract the NZE index for each entry from a sparse matrix. More...
template<typename T >
const std::vector< int >	csr_extract_v (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
	Extract the column indexes for non-zero value from a sparse matrix. More...
template<typename T , require_eigen_dense_base_t< T > * = nullptr>
const std::vector< int >	csr_extract_v (const T &A)
	Extract the column indexes for non-zero values from a dense matrix by converting to sparse and calling the sparse matrix extractor. More...
template<typename T >
const Eigen::Matrix< T, Eigen::Dynamic, 1 >	csr_extract_w (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
template<typename T , require_eigen_dense_base_t< T > * = nullptr>
const Eigen::Matrix< scalar_type_t< T >, Eigen::Dynamic, 1 >	csr_extract_w (const T &A)
template<typename T1 , typename T2 , require_all_not_rev_matrix_t< T1, T2 > * = nullptr>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, 1 >	csr_matrix_times_vector (int m, int n, const T1 &w, const std::vector< int > &v, const std::vector< int > &u, const T2 &b)
template<typename T >
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	csr_to_dense_matrix (int m, int n, const T &w, const std::vector< int > &v, const std::vector< int > &u)
	Construct a dense Eigen matrix from the CSR format components. More...
int	csr_u_to_z (const std::vector< int > &u, int i)
	Return the z vector computed from the specified u vector at the index for the z vector. More...
template<typename T >
std::vector< T >	cumulative_sum (const std::vector< T > &x)
	Return the cumulative sum of the specified vector. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
auto	cumulative_sum (const EigVec &m)
	Return the cumulative sum of the specified vector. More...
template<typename T , require_eigen_vt< std::is_arithmetic, T > * = nullptr>
value_type_t< T >	determinant (const T &m)
	Returns the determinant of the specified square matrix. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
Eigen::Matrix< value_type_t< EigVec >, Eigen::Dynamic, Eigen::Dynamic >	diag_matrix (const EigVec &v)
	Return a square diagonal matrix with the specified vector of coefficients as the diagonal values. More...
template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_eigen_vector_t< T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>
auto	diag_post_multiply (const T1 &m1, const T2 &m2)
	Return the product of the matrix and the diagonal matrix formed from the vector or row_vector. More...
template<typename T1 , typename T2 , require_eigen_vector_t< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>
auto	diag_pre_multiply (const T1 &m1, const T2 &m2)
	Return the product of the diagonal matrix formed from the vector or row_vector and a matrix. More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	diagonal (const T &m)
	Return a column vector of the diagonal elements of the specified matrix. More...
double	digamma (double x)
	Return the derivative of the log gamma function at the specified value. More...
template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	digamma (const T &x)
	Vectorized version of digamma(). More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
void	dims (const T &x, std::vector< int > &result)
	Pushes dimensions of given argument into given result vector. More...
template<typename T , typename Alloc >
void	dims (const std::vector< T, Alloc > &x, std::vector< int > &result)
	Pushes dimensions of given argument into given result vector. More...
template<typename T >
std::vector< int >	dims (const T &x)
	Determines dimensions of an argument. More...
template<typename T1 , typename T2 , require_all_stan_scalar_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	distance (const T1 &x1, const T2 &x2)
	Returns the distance between two scalars. More...
template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr>
return_type_t< Scal1, Scal2 >	divide (const Scal1 &x, const Scal2 &y)
	Return the division of the first scalar by the second scalar. More...
int	divide (int x, int y)
template<typename T1 , typename T2 , require_any_eigen_t< T1, T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>
auto	divide (const T1 &m, const T2 &c)
	Return matrix divided by scalar. More...
template<typename T_x , typename T_v >
std::vector< Eigen::Matrix< return_type_t< T_x, T_v, double >, Eigen::Dynamic, 1 > >	divide_columns (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const std::vector< T_v > &vec)
	Takes Stan data type vector[n] x[D] and divides column vector in x element-wise by the values in vec. More...
double	dot (const std::vector< double > &x, const std::vector< double > &y)
template<typename Vec1 , typename Vec2 , require_all_eigen_vector_t< Vec1, Vec2 > * = nullptr, require_not_var_t< return_type_t< Vec1, Vec2 >> * = nullptr>
auto	dot_product (const Vec1 &v1, const Vec2 &v2)
	Returns the dot product of the specified vectors. More...
template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr, require_all_not_var_t< Scalar1, Scalar2 > * = nullptr>
auto	dot_product (const Scalar1 *v1, const Scalar2 *v2, size_t length)
	Returns the dot product of the specified arrays. More...
template<typename Scalar1 , typename Scalar2 , typename Alloc1 , typename Alloc2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>
return_type_t< Scalar1, Scalar2 >	dot_product (const std::vector< Scalar1, Alloc1 > &v1, const std::vector< Scalar2, Alloc2 > &v2)
	Returns the dot product of the specified arrays. More...
double	dot_self (const std::vector< double > &x)
template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_var, T > * = nullptr>
value_type_t< T >	dot_self (const T &v)
	Returns squared norm of a vector or matrix. More...
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator< (const T_a &a, const T_b &b)
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator<= (const T_a &a, const T_b &b)
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator> (const T_a &a, const T_b &b)
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator>= (const T_a &a, const T_b &b)
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator== (const T_a &a, const T_b &b)
template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>
auto	operator!= (const T_a &a, const T_b &b)
template<typename T >
Eigen::Matrix< std::complex< T >,-1, 1 >	eigenvalues (const Eigen::Matrix< T,-1,-1 > &m)
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, 1 >	eigenvalues_sym (const EigMat &m)
	Return the eigenvalues of the specified symmetric matrix in descending order of magnitude. More...
template<typename T >
Eigen::Matrix< std::complex< T >,-1,-1 >	eigenvectors (const Eigen::Matrix< T,-1,-1 > &m)
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	eigenvectors_sym (const EigMat &m)
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>
auto	elt_divide (const Mat1 &m1, const Mat2 &m2)
	Return the elementwise division of the specified matrices. More...
template<typename Mat , typename Scal , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr>
auto	elt_divide (const Mat &m, Scal s)
	Return the elementwise division of the specified matrix by the specified scalar. More...
template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr>
auto	elt_divide (Scal s, const Mat &m)
	Return the elementwise division of the specified scalar by the specified matrix. More...
template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr>
auto	elt_divide (Scal1 s1, Scal2 s2)
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>
auto	elt_multiply (const Mat1 &m1, const Mat2 &m2)
	Return the elementwise multiplication of the specified matrices. More...
template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>
auto	elt_multiply (const Scalar1 &a, const Scalar2 &b)
	Return the elementwise multiplication of the specified scalars. More...
template<typename T1 , typename T2 , require_any_matrix_t< T1, T2 > * = nullptr, require_any_stan_scalar_t< T1, T2 > * = nullptr>
auto	elt_multiply (const T1 &A, const T2 &B)
	Return specified matrix multiplied by specified scalar. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	erf (const T &x)
	Returns the elementwise erf() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	erfc (const T &x)
	Returns the elementwise erfc() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T , require_same_t< std::decay_t< T >, plain_type_t< T >> * = nullptr>
T	eval (T &&arg)
	Inputs which have a plain_type equal to the own time are forwarded unmodified (for Eigen expressions these types are different) More...
template<typename T , require_not_same_t< std::decay_t< T >, plain_type_t< T >> * = nullptr>
decltype(auto)	eval (const T &arg)
	Inputs which have a plain_type different from their own type are Eval'd (this catches Eigen expressions) More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>
auto	exp (const Container &x)
	Return the elementwise exp() of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	exp (const Container &x)
	Version of exp() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	exp2 (const T &x)
	Return the elementwise exp2() of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	expm1 (const T &x)
	Return the elementwise expm1() of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T >
T	F32 (const T &a1, const T &a2, const T &a3, const T &b1, const T &b2, const T &z, double precision=1e-6, int max_steps=1e5)
	Hypergeometric function (3F2). More...
template<typename T , require_arithmetic_t< T > * = nullptr>
auto	fabs (T x)
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_stan_scalar_t< Container > * = nullptr>
auto	fabs (const Container &x)
	Returns the elementwise fabs() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	fabs (const Container &x)
	Version of fabs() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T_Sigma , typename T_CPCs , typename T_sds , require_eigen_t< T_Sigma > * = nullptr, require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_all_vt_same< T_Sigma, T_CPCs, T_sds > * = nullptr>
bool	factor_cov_matrix (const T_Sigma &Sigma, T_CPCs &&CPCs, T_sds &&sds)
	This function is intended to make starting values, given a covariance matrix Sigma. More...
template<typename T_U , typename T_CPCs , require_eigen_t< T_U > * = nullptr, require_eigen_vector_t< T_CPCs > * = nullptr, require_vt_same< T_U, T_CPCs > * = nullptr>
void	factor_U (const T_U &U, T_CPCs &&CPCs)
	This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
return_type_t< T >	falling_factorial (const T &x, int n)
	Return the falling factorial function evaluated at the inputs. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>
auto	falling_factorial (const T1 &a, const T2 &b)
	Enables the vectorised application of the falling factorial function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
double	fdim (T1 x, T2 y)
	Return the positive difference of the specified values (C++11). More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	fdim (const T1 &a, const T2 &b)
	Enables the vectorised application of the fdim function, when the first and/or second arguments are containers. More...
template<typename EigMat , typename S , require_eigen_t< EigMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>
void	fill (EigMat &x, const S &y)
	Fill the specified container with the specified value. More...
template<typename T , typename S , require_t< std::is_assignable< std::decay_t< T > &, std::decay_t< S >>> * = nullptr>
void	fill (T &x, S &&y)
	Fill the specified container with the specified value. More...
template<typename Vec , typename S , require_std_vector_t< Vec > * = nullptr>
void	fill (Vec &x, S &&y)
	Fill the specified container with the specified value. More...
double	finite_diff_stepsize (double u)
	Return the stepsize for finite difference evaluations at the specified scalar. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>
auto	floor (const Container &x)
	Returns the elementwise floor() of the input, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>
auto	floor (const Container &x)
	Version of floor() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T1 , typename T2 , typename T3 , require_all_arithmetic_t< T1, T2, T3 > * = nullptr>
double	fma (T1 x, T2 y, T3 z)
	Return the product of the first two arguments plus the third argument. More...
template<typename T1 , typename T2 , typename T3 , require_any_matrix_t< T1, T2, T3 > * = nullptr, require_not_var_t< return_type_t< T1, T2, T3 >> * = nullptr>
auto	fma (T1 &&x, T2 &&y, T3 &&z)
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
double	fmax (T1 x, T2 y)
	Return the greater of the two specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	fmax (const T1 &a, const T2 &b)
	Enables the vectorised application of the fmax function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
double	fmin (T1 x, T2 y)
	Return the lesser of the two specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	fmin (const T1 &a, const T2 &b)
	Enables the vectorised application of the fmin function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	fmod (const T1 &a, const T2 &b)
	Enables the vectorised application of the fmod function, when the first and/or second arguments are containers. More...
double	gamma_p (double z, double a)
	Return the value of the normalized, lower-incomplete gamma function applied to the specified argument. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	gamma_p (const T1 &a, const T2 &b)
	Enables the vectorised application of the gamma_p function, when the first and/or second arguments are containers. More...
double	gamma_q (double x, double a)
	\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \] More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	gamma_q (const T1 &a, const T2 &b)
	Enables the vectorised application of the gamma_q function, when the first and/or second arguments are containers. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::ColsAtCompileTime, EigMat::RowsAtCompileTime >	generalized_inverse (const EigMat &G)
	Returns the Moore-Penrose generalized inverse of the specified matrix. More...
template<typename T >
const T &	get_base1 (const std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one index. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vector_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, 1, Eigen::Dynamic >	get_base1 (const EigMat &x, size_t m, const char *error_msg, size_t idx)
	Return a copy of the row of the specified matrix at the specified base-one row index. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
const value_type_t< EigMat > &	get_base1 (const EigMat &x, size_t m, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
const value_type_t< EigVec > &	get_base1 (const EigVec &x, size_t m, const char *error_msg, size_t idx)
	Return a reference to the value of the specified Eigen vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
Eigen::Block< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > >	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, const char *error_msg, size_t idx)
	Return a copy of the row of the specified vector at the specified base-one row index. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t m, const char *error_msg, size_t idx)
	Return a reference to the value of the specified column vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, 1, Eigen::Dynamic > &x, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified row vector at the specified base-one index. More...
template<typename T >
T	get_imag (const std::complex< T > &z)
	Return the imaginary component of the complex argument. More...
template<typename T_lp , typename T_lp_accum >
return_type_t< T_lp, T_lp_accum >	get_lp (const T_lp &lp, const accumulator< T_lp_accum > &lp_accum)
template<typename T >
T	get_real (const std::complex< T > &z)
	Return the real component of the complex argument. More...
template<typename T_x , typename T_sigma >
Eigen::Matrix< return_type_t< T_x, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const T_sigma &sigma)
	Returns a dot product covariance matrix. More...
template<typename T_x , typename T_sigma >
Eigen::Matrix< return_type_t< T_x, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< T_x > &x, const T_sigma &sigma)
	Returns a dot product covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_sigma >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &x2, const T_sigma &sigma)
	Returns a dot product covariance matrix of differing x's. More...
template<typename T_x1 , typename T_x2 , typename T_sigma >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma)
	Returns a dot product covariance matrix of differing x's. More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_sigma &sigma, const std::vector< T_l > &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern exponential covariance Matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern exponential covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern exponential cross covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern exponential cross covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 3/2 covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 3/2 covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 3/2 cross covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 3/2 cross covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 5/2 covariance matrix with one input vector. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 5/2 covariance matrix with one input vector with automatic relevance determination (ARD). More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 5/2 covariance matrix with two different input vectors. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 5/2 covariance matrix with two input vectors with automatic relevance determination (ARD). More...
template<typename T_x , typename T_sigma , typename T_l , typename T_p >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l, T_p >, Eigen::Dynamic, Eigen::Dynamic >	gp_periodic_cov (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &l, const T_p &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \). More...
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l , typename T_p >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l, T_p >, Eigen::Dynamic, Eigen::Dynamic >	gp_periodic_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &l, const T_p &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using inputs \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \). More...
template<typename T >
void	grad_2F1 (T &g_a1, T &g_b1, const T &a1, const T &a2, const T &b1, const T &z, double precision=1e-14, int max_steps=1e6)
	Gradients of the hypergeometric function, 2F1. More...
template<typename T >
void	grad_F32 (T *g, const T &a1, const T &a2, const T &a3, const T &b1, const T &b2, const T &z, const T &precision=1e-6, int max_steps=1e5)
	Gradients of the hypergeometric function, 3F2. More...
void	grad_inc_beta (double &g1, double &g2, double a, double b, double z)
template<typename T >
void	grad_reg_inc_beta (T &g1, T &g2, const T &a, const T &b, const T &z, const T &digammaA, const T &digammaB, const T &digammaSum, const T &betaAB)
	Computes the gradients of the regularized incomplete beta function. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	grad_reg_inc_gamma (T1 a, T2 z, T1 g, T1 dig, double precision=1e-6, int max_steps=1e5)
	Gradient of the regularized incomplete gamma functions igamma(a, z) More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	grad_reg_lower_inc_gamma (const T1 &a, const T2 &z, double precision=1e-10, int max_steps=1e5)
	Computes the gradient of the lower regularized incomplete gamma function. More...
template<typename T , require_vector_t< T > * = nullptr>
auto	head (const T &v, size_t n)
	Return the specified number of elements as a vector or row vector (same as input) from the front of the specified vector or row vector. More...
template<typename T >
std::vector< T >	head (const std::vector< T > &sv, size_t n)
	Return the specified number of elements as a standard vector from the front of the specified standard vector. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
double	hypot (T1 x, T2 y)
	Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
auto	hypot (const T1 &a, const T2 &b)
	Enables the vectorised application of the hypot function, when the first and/or second arguments are containers. More...
template<typename T >
std::complex< T >	i_times (const std::complex< T > &z)
	Return the specified complex number multiplied by i. More...
template<typename T >
std::complex< T >	neg_i_times (const std::complex< T > &z)
	Return the specified complex number multiplied by -i. More...
template<typename V >
std::complex< V >	complex_negate (const std::complex< V > &z)
	Return the complex negation of the specified complex argument. More...
template<bool Jacobian = false, typename T , typename... Types, require_all_not_var_matrix_t< T, Types... > * = nullptr>
auto	identity_constrain (T &&x, Types &&...)
	Returns the result of applying the identity constraint transform to the input. More...
template<typename T , typename... Types, require_all_not_var_matrix_t< T, Types... > * = nullptr>
auto	identity_free (T &&x, Types &&...)
	Returns the result of applying the inverse of the identity constraint transform to the input. More...
template<typename T_true , typename T_false >
return_type_t< T_true, T_false >	if_else (const bool c, const T_true y_true, const T_false y_false)
	Return the second argument if the first argument is true and otherwise return the second argument. More...
template<typename T , require_autodiff_t< T > >
T	imag (const std::complex< T > &z)
	Return the imaginary component of the complex argument. More...
double	inc_beta (double a, double b, double x)
	The normalized incomplete beta function of a, b, with outcome x. More...
template<typename T >
T	inc_beta_dda (T a, T b, T z, T digamma_a, T digamma_ab)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a. More...
template<typename T >
T	inc_beta_ddb (T a, T b, T z, T digamma_b, T digamma_ab)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b. More...
template<typename T >
T	inc_beta_ddz (T a, T b, T z)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z. More...
template<>
double	inc_beta_ddz (double a, double b, double z)
template<typename T , typename V , require_all_stan_scalar_t< T, V > * = nullptr, require_convertible_t< V, T > * = nullptr>
void	initialize (T &x, V v)
template<typename T , int R, int C, typename V >
void	initialize (Eigen::Matrix< T, R, C > &x, const V &v)
template<typename T , typename V >
void	initialize (std::vector< T > &x, const V &v)
template<typename EigMat , typename S , require_eigen_t< EigMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>
void	initialize_fill (EigMat &x, const S &y)
	Fill the specified container with the specified value. More...
template<typename T , typename S , require_t< std::is_assignable< std::decay_t< T > &, std::decay_t< S >>> * = nullptr>
void	initialize_fill (T &x, S &&y)
	Fill the specified container with the specified value. More...
template<typename Vec , typename S , require_std_vector_t< Vec > * = nullptr>
void	initialize_fill (Vec &x, S &&y)
	Fill the specified container with the specified value. More...
template<typename T >
int	int_step (const T &y)
	The integer step, or Heaviside, function. More...
template<typename T , require_not_container_st< std::is_arithmetic, T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	inv (const T &x)
	Return the elementwise 1.0 / x of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	inv (const Container &x)
	Version of inv() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
double	inv_cloglog (double x)
	The inverse complementary log-log function. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	inv_cloglog (const Container &x)
	Vectorized version of inv_cloglog(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	inv_cloglog (const Container &x)
	Version of inv_cloglog() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
double	inv_logit (double a)
	Returns the inverse logit function applied to the argument. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	inv_logit (const T &x)
	Vectorized version of inv_logit(). More...
double	inv_Phi (double p)
	Return the value of the inverse standard normal cumulative distribution function at the specified argument. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	inv_Phi (const T &x)
	Vectorized version of inv_Phi(). More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
auto	inv_sqrt (T x)
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_stan_scalar_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	inv_sqrt (const Container &x)
	Return the elementwise 1 / sqrt(x)} of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_not_var_matrix_t< Container > * = nullptr, require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	inv_sqrt (const Container &x)
	Version of inv_sqrt() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	inv_square (const Container &x)
	Returns 1 / square(x). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	inv_square (const Container &x)
	Version of inv_square() that accepts Eigen Matrix/Array objects or expressions. More...
template<typename Vector , require_vector_t< Vector > * = nullptr>
void	inverse_softmax (const Vector &simplex, Vector &y)
	Writes the inverse softmax of the simplex argument into the second argument. More...
template<typename EigMat >
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	inverse_spd (const EigMat &m)
	Returns the inverse of the specified symmetric, pos/neg-definite matrix. More...
template<typename T >
bool	is_any_nan (const T &x)
	Returns true if the input is NaN and false otherwise. More...
template<typename T , typename... Ts>
bool	is_any_nan (const T &x, const Ts &...xs)
	Returns true if any input is NaN and false otherwise. More...
bool	is_inf (double x)
	Returns true if the input is infinite and false otherwise. More...
template<typename T >
bool	is_integer (T x)
	Returns true if the input is an integer and false otherwise. More...
template<typename T , typename = require_arithmetic_t<T>>
bool	is_nan (T x)
	Returns true if the input is NaN and false otherwise. More...
template<typename T , typename = require_eigen_t<T>>
bool	is_nan (const T &x)
template<typename T >
bool	is_nonpositive_integer (T x)
	Returns true if the input is a nonpositive integer and false otherwise. More...
template<typename T >
bool	is_uninitialized (T x)
	Returns true if the specified variable is uninitialized. More...
template<typename ADType , require_autodiff_t< ADType > * = nullptr>
bool	isfinite (ADType &&v)
	Return true if specified argument is finite (not infinite and not not-a-number). More...
template<typename T , typename = require_autodiff_t<T>>
bool	isinf (const T &v)
	Return true if specified argument is infinite (positive or negative). More...
template<typename T , typename = require_autodiff_t<T>>
bool	isnan (const T &x)
	Return true if specified argument is not-a-number. More...
template<typename ADType , require_autodiff_t< ADType > * = nullptr>
bool	isnormal (ADType &&v)
	Return true if specified argument is normal. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
double	lambert_w0 (const T &x)
	Compute the Lambert W function on W0 branch for a value x. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
double	lambert_wm1 (const T &x)
	Compute the Lambert W function on W-1 branch for a value x. More...
template<typename T , require_not_stan_scalar_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	lambert_w0 (const T &x)
	Vectorized version of lambert_w0(). More...
template<typename T , require_not_stan_scalar_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	lambert_wm1 (const T &x)
	Vectorized version of lambert_wm1(). More...
template<typename T , typename L , require_all_stan_scalar_t< T, L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>
auto	lb_constrain (const T &x, const L &lb)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T , typename L , require_eigen_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>
auto	lb_constrain (T &&x, L &&lb)
	Specialization of lb_constrain to apply a scalar lower bound elementwise to each input. More...
template<typename T , typename L , require_eigen_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>
auto	lb_constrain (const T &x, const L &lb, return_type_t< T, L > &lp)
	Specialization of lb_constrain to apply a scalar lower bound elementwise to each input. More...
template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>
auto	lb_constrain (const std::vector< T > &x, const L &lb)
	Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element. More...
template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>
auto	lb_constrain (const std::vector< T > &x, const L &lb, return_type_t< T, L > &lp)
	Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element. More...
template<typename T , typename L >
auto	lb_constrain (const std::vector< T > &x, const std::vector< L > &lb)
	Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element. More...
template<typename T , typename L >
auto	lb_constrain (const std::vector< T > &x, const std::vector< L > &lb, return_type_t< T, L > &lp)
	Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element. More...
template<bool Jacobian, typename T , typename L >
auto	lb_constrain (const T &x, const L &lb, return_type_t< T, L > &lp)
	Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element. More...
template<typename T , typename L , require_not_std_vector_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr>
auto	lb_free (T &&y, L &&lb)
	Return the unconstrained value that produces the specified lower-bound constrained value. More...
template<typename T , typename L , require_all_eigen_t< T, L > * = nullptr>
auto	lb_free (T &&y, L &&lb)
	Return the free matrix that corresponds to the specified lower-bounded matrix with respect to the specified lower bound. More...
template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>
auto	lb_free (const std::vector< T > y, const L &lb)
	Return the free variable that corresponds to the specified lower-bounded variable with respect to the specified lower bound. More...
template<typename T , typename L >
auto	lb_free (const std::vector< T > y, const std::vector< L > &lb)
	Return the free variable that corresponds to the specified lower-bounded variable with respect to the specified lower bound. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	lbeta (const T1 a, const T2 b)
	Return the log of the beta function applied to the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	lbeta (const T1 &a, const T2 &b)
	Enables the vectorised application of the lbeta function, when the first and/or second arguments are containers. More...
template<typename T1 , require_arithmetic_t< T1 > * = nullptr>
double	ldexp (T1 a, int b)
	Returns the product of a (the significand) and 2 to power b (the exponent). More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>
auto	ldexp (const T1 &a, const T2 &b)
	Enables the vectorised application of the ldexp function, when the first and/or second arguments are containers. More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	make_ldlt_factor (const T &A)
	Make an LDLT_factor with matrix type plain_type_t<T> More...
double	lgamma (double x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
double	lgamma (int x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	lgamma (const T &x)
	Vectorized version of lgamma(). More...
template<typename T >
return_type_t< T >	lgamma_stirling (const T x)
	Return the Stirling approximation to the lgamma function. More...
template<typename T >
return_type_t< T >	lgamma_stirling_diff (const T x)
	Return the difference between log of the gamma function and its Stirling approximation. More...
std::vector< int >	linspaced_array (int K, double low, double high)
	Return an array of linearly spaced elements. More...
std::vector< int >	linspaced_int_array (int K, int low, int high)
	Return an array of linearly spaced integers. More...
auto	linspaced_row_vector (int K, double low, double high)
	Return a row vector of linearly spaced elements. More...
auto	linspaced_vector (int K, double low, double high)
	Return a vector of linearly spaced elements. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
return_type_t< T >	lmgamma (int k, T x)
	Return the natural logarithm of the multivariate gamma function with the specified dimensions and argument. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	lmgamma (const T1 &a, const T2 &b)
	Enables the vectorised application of the natural log of the multivariate gamma function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	lmultiply (const T1 a, const T2 b)
	Return the first argument times the log of the second argument. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>
auto	lmultiply (const T1 &a, const T2 &b)
	Return the result of applying lmultiply to the arguments elementwise, with broadcasting if one of the arguments is a scalar. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	log (const Container &x)
	Return the elementwise natural log of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	log (const Container &x)
	Version of log() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename Container , require_not_var_matrix_t< Container > * = nullptr, require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	log10 (const Container &x)
	Vectorized version of log10(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	log10 (const Container &x)
	Version of log10() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
double	log1m (double x)
	Return the natural logarithm of one minus the specified value. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	log1m (const T &x)
	Vectorized version of log1m(). More...
double	log1m_exp (double a)
	Calculates the natural logarithm of one minus the exponential of the specified value without overflow,. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	log1m_exp (const T &x)
	Vectorized version of log1m_exp(). More...
double	log1m_inv_logit (double u)
	Returns the natural logarithm of 1 minus the inverse logit of the specified argument. More...
double	log1m_inv_logit (int u)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
apply_scalar_unary< log1m_inv_logit_fun, T >::return_t	log1m_inv_logit (const T &x)
	Return the elementwise application of log1m_inv_logit() to specified argument container. More...
double	log1p (double x)
	Return the natural logarithm of one plus the specified value. More...
double	log1p (int x)
	Return the natural logarithm of one plus the specified argument. More...
template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	log1p (const T &x)
	Return the elementwise application of log1p() to specified argument container. More...
double	log1p_exp (double a)
	Calculates the log of 1 plus the exponential of the specified value without overflow. More...
template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	log1p_exp (const T &x)
	Vectorized version of log1p_exp(). More...
double	log2 ()
	Return natural logarithm of two. More...
template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	log2 (const T &x)
	Return the elementwise application of log2() to specified argument container. More...
template<typename T , require_not_rev_matrix_t< T > * = nullptr>
value_type_t< T >	log_determinant_ldlt (LDLT_factor< T > &A)
	Returns log(abs(det(A))) given a LDLT_factor of A. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>
value_type_t< EigMat >	log_determinant_spd (const EigMat &m)
	Returns the log absolute determinant of the specified square matrix. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	log_diff_exp (const T1 x, const T2 y)
	The natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	log_diff_exp (const T1 &a, const T2 &b)
	Enables the vectorised application of the log_diff_exp function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	log_falling_factorial (const T1 x, const T2 n)
	Return the natural log of the falling factorial of the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	log_falling_factorial (const T1 &a, const T2 &b)
	Enables the vectorised application of the log_falling_factorial function, when the first and/or second arguments are containers. More...
double	log_inv_logit (double u)
	Returns the natural logarithm of the inverse logit of the specified argument. More...
double	log_inv_logit (int u)
	Returns the natural logarithm of the inverse logit of the specified argument. More...
template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	log_inv_logit (const T &x)
	Return the elementwise application of log_inv_logit() to specified argument container. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	log_inv_logit_diff (const T1 &x, const T2 &y)
	Returns the natural logarithm of the difference of the inverse logits of the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	log_inv_logit_diff (const T1 &a, const T2 &b)
	Enables the vectorised application of the log_inv_logit_diff function, when the first and/or second arguments are containers. More...
template<typename T_theta , typename T_lambda1 , typename T_lambda2 , require_all_arithmetic_t< T_theta, T_lambda1, T_lambda2 > * = nullptr>
double	log_mix (T_theta theta, T_lambda1 lambda1, T_lambda2 lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
template<typename T_theta , typename T_lam , require_any_vector_t< T_theta, T_lam > * = nullptr>
return_type_t< T_theta, T_lam >	log_mix (const T_theta &theta, const T_lam &lambda)
	Return the log mixture density with specified mixing proportions and log densities. More...
template<typename T_theta , typename T_lam , require_vector_t< T_lam > * = nullptr>
return_type_t< T_theta, std::vector< T_lam > >	log_mix (const T_theta &theta, const std::vector< T_lam > &lambda)
	Return the log mixture density given specified mixing proportions and array of log density vectors. More...
template<typename T1 , typename T2 , require_all_stan_scalar_t< T1, T2 > * = nullptr>
return_type_t< T1, T2, double >	log_modified_bessel_first_kind (const T1 v, const T2 z)
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	log_modified_bessel_first_kind (const T1 &a, const T2 &b)
	Enables the vectorised application of the log_modified_bessel_first_kind function, when the first and/or second arguments are containers. More...
template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	log_rising_factorial (const T1 &x, const T2 &n)
	Return the natural logarithm of the rising factorial from the first specified argument to the second. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	log_rising_factorial (const T1 &a, const T2 &b)
	Enables the vectorised application of the log_rising_factorial function, when the first and/or second arguments are containers. More...
template<typename Container , require_st_arithmetic< Container > * = nullptr, require_container_t< Container > * = nullptr>
auto	log_softmax (const Container &x)
	Return the natural logarithm of the softmax of the specified vector. More...
template<typename T1 , typename T2 , require_all_not_st_var< T1, T2 > * = nullptr>
return_type_t< T1, T2 >	log_sum_exp (const T2 &a, const T1 &b)
	Calculates the log sum of exponentials without overflow. More...
template<typename T , typename = require_autodiff_t<T>>
double	logb (const T &x)
	Returns the value of the unbiased radix-independent exponent from the floating-point argument. More...
template<typename T1 , typename T2 >
int	logical_and (const T1 x1, const T2 x2)
	The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise. More...
template<typename T1 , typename T2 >
int	logical_eq (const T1 x1, const T2 x2)
	Return 1 if the first argument is equal to the second. More...
template<typename T1 , typename T2 >
bool	logical_gt (const T1 x1, const T2 x2)
	Return 1 if the first argument is strictly greater than the second. More...
template<typename T1 , typename T2 >
bool	logical_gte (const T1 x1, const T2 x2)
	Return 1 if the first argument is greater than or equal to the second. More...
template<typename T1 , typename T2 >
bool	logical_lt (T1 x1, T2 x2)
	Return 1 if the first argument is strictly less than the second. More...
template<typename T1 , typename T2 >
bool	logical_lte (const T1 x1, const T2 x2)
	Return 1 if the first argument is less than or equal to the second. More...
template<typename T >
int	logical_negation (const T &x)
	The logical negation function which returns one if the input is equal to zero and zero otherwise. More...
template<typename T1 , typename T2 >
int	logical_neq (const T1 x1, const T2 x2)
	Return 1 if the first argument is unequal to the second. More...
template<typename T1 , typename T2 >
int	logical_or (T1 x1, T2 x2)
	The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise. More...
double	logit (double u)
	Return the log odds of the argument. More...
double	logit (int u)
	Return the log odds of the argument. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	logit (const Container &x)
	Return the elementwise application of logit() to specified argument container. More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	logit (const Container &x)
	Version of logit() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T , typename L , typename U , require_all_stan_scalar_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (T &&x, L &&lb, U &&ub)
	Return the lower and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds. More...
template<typename T , typename L , typename U , require_all_stan_scalar_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (T &&x, L &&lb, U &&ub, return_type_t< T, L, U > &lp)
	Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log density with the log absolute Jacobian determinant. More...
template<typename T , typename L , typename U , require_eigen_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (const T &x, const L &lb, const U &ub)
	Overload for Eigen matrix and scalar bounds. More...
template<typename T , typename L , typename U , require_eigen_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (const T &x, const L &lb, const U &ub, return_type_t< T, L, U > &lp)
	Overload for Eigen matrix and scalar bounds plus lp. More...
template<typename T , typename L , typename U , require_all_eigen_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (const T &x, const L &lb, const U &ub)
	Overload for Eigen matrix and matrix bounds. More...
template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const L &lb, const U &ub)
	Overload for array of x and non-array lb and ub. More...
template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const L &lb, const U &ub, return_type_t< T, L, U > &lp)
	Overload for array of x and non-array lb and ub with lp. More...
template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const L &lb, const std::vector< U > &ub)
	Overload for array of x and ub and non-array lb. More...
template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const L &lb, const std::vector< U > &ub, return_type_t< T, L, U > &lp)
	Overload for array of x and ub and non-array lb with lp. More...
template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const std::vector< L > &lb, const U &ub)
	Overload for array of x and lb and non-array ub. More...
template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>
auto	lub_constrain (const std::vector< T > &x, const std::vector< L > &lb, const U &ub, return_type_t< T, L, U > &lp)
	Overload for array of x and lb and non-array ub with lp. More...
template<typename T , typename L , typename U >
auto	lub_constrain (const std::vector< T > &x, const std::vector< L > &lb, const std::vector< U > &ub)
	Overload for array of x, lb, and ub with lp. More...
template<typename T , typename L , typename U >
auto	lub_constrain (const std::vector< T > &x, const std::vector< L > &lb, const std::vector< U > &ub, return_type_t< T, L, U > &lp)
	Overload for array of x, lb, and ub. More...
template<bool Jacobian, typename T , typename L , typename U >
auto	lub_constrain (const T &x, const L &lb, const U &ub, return_type_t< T, L, U > &lp)
	Return the lower and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds. More...
template<typename T >
Eigen::Array< T, Eigen::Dynamic, 1 >	make_nu (const T &eta, size_t K)
	Return the degrees of freedom for the t distribution that corresponds to the shape parameter in the LKJ distribution. More...
template<typename T , typename = require_eigen_t<T>>
plain_type_t< T >	matrix_exp (const T &A_in)
	Return the matrix exponential of the input matrix. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	matrix_exp_2x2 (const EigMat &A)
	Return the matrix exponential of a 2x2 matrix. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_st_same< double, EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< double, Eigen::Dynamic, EigMat2::ColsAtCompileTime >	matrix_exp_multiply (const EigMat1 &A, const EigMat2 &B)
	Return product of exp(A) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime >	matrix_exp_pade (const EigMat &arg)
	Computes the matrix exponential, using a Pade approximation, coupled with scaling and squaring. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime >	matrix_power (const EigMat &M, const int n)
	Returns the nth power of the specific matrix. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime >	operator^ (const EigMat &M, const int n)
template<typename T , require_container_t< T > * = nullptr>
value_type_t< T >	max (const T &m)
	Returns the maximum coefficient in the specified matrix, vector, row vector or std vector. More...
template<typename T1 , typename... Ts>
size_t	max_size (const T1 &x1, const Ts &...xs)
	Calculate the size of the largest input. More...
template<typename T1 , typename... Ts>
size_t	max_size_mvt (const T1 &x1, const Ts &...xs)
	Calculate the size of the largest multivariate input. More...
template<typename T1 , typename T2 , require_all_eigen_vt< std::is_arithmetic, T1, T2 > * = nullptr>
Eigen::Matrix< return_type_t< T1, T2 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left (const T1 &A, const T2 &b)
	Returns the solution of the system Ax=b. More...
template<typename T , typename EigMat , require_eigen_t< EigMat > * = nullptr, require_all_not_st_var< T, EigMat > * = nullptr, require_any_not_t< std::is_arithmetic< value_type_t< T >>, is_fvar< value_type_t< EigMat >>> * = nullptr>
Eigen::Matrix< return_type_t< T, EigMat >, Eigen::Dynamic, EigMat::ColsAtCompileTime >	mdivide_left_ldlt (LDLT_factor< T > &A, const EigMat &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_var< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< return_type_t< EigMat1, EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_left_spd (const EigMat1 &A, const EigMat2 &b)
	Returns the solution of the system Ax=b where A is symmetric positive definite. More...
template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_all_not_eigen_vt< is_var, T1, T2 > * = nullptr>
Eigen::Matrix< return_type_t< T1, T2 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri (const T1 &A, const T2 &b)
	Returns the solution of the system Ax=b when A is triangular. More...
template<Eigen::UpLoType TriView, typename T , require_eigen_t< T > * = nullptr>
plain_type_t< T >	mdivide_left_tri (const T &A)
	Returns the solution of the system Ax=b when A is triangular and b=I. More...
template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_all_not_eigen_vt< is_fvar, T1, T2 > * = nullptr>
Eigen::Matrix< return_type_t< T1, T2 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri_low (const T1 &A, const T2 &b)
	Return the result of left dividing the second argument by the first argument. More...
template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_fvar, T > * = nullptr>
plain_type_t< T >	mdivide_left_tri_low (const T &A)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_fvar< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< return_type_t< EigMat1, EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right (const EigMat1 &b, const EigMat2 &A)
	Returns the solution of the system xA=b. More...
template<typename EigMat , typename T , require_all_matrix_t< EigMat, T > * = nullptr, require_any_not_st_arithmetic< EigMat, T > * = nullptr>
auto	mdivide_right_ldlt (const EigMat &b, LDLT_factor< T > &A)
	Returns the solution of the system xA=b given an LDLT_factor of A. More...
template<typename EigMat , typename T , require_all_matrix_t< EigMat, T > * = nullptr, require_all_st_arithmetic< EigMat, T > * = nullptr>
Eigen::Matrix< double, EigMat::RowsAtCompileTime, T::ColsAtCompileTime >	mdivide_right_ldlt (const EigMat &b, LDLT_factor< T > &A)
	Returns the solution of the system xA=b given an LDLT_factor of A. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< return_type_t< EigMat1, EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right_spd (const EigMat1 &b, const EigMat2 &A)
	Returns the solution of the system xA=b where A is symmetric positive definite. More...
template<Eigen::UpLoType TriView, typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< return_type_t< EigMat1, EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right_tri (const EigMat1 &b, const EigMat2 &A)
	Returns the solution of the system xA=b when A is triangular. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_fvar< EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< return_type_t< EigMat1, EigMat2 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_right_tri_low (const EigMat1 &b, const EigMat2 &A)
	Returns the solution of the system x tri(A) = b when tri(A) is a lower triangular view of the matrix A. More...
template<typename T , require_container_t< T > * = nullptr>
return_type_t< T >	mean (const T &m)
	Returns the sample mean (i.e., average) of the coefficients in the specified std vector, vector, row vector, or matrix. More...
template<typename T , require_container_t< T > * = nullptr>
value_type_t< T >	min (const T &m)
	Returns the minimum coefficient in the specified matrix, vector, row vector or std vector. More...
template<typename T >
auto	minus (const T &x)
	Returns the negation of the specified scalar or matrix. More...
template<typename T2 , require_arithmetic_t< T2 > * = nullptr>
T2	modified_bessel_first_kind (int v, const T2 z)
	\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \] More...
double	modified_bessel_first_kind (int v, int z)
	This function exists because when z is of type integer, cyl_bessel_i(v, z) returns an integer. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	modified_bessel_first_kind (const T1 &a, const T2 &b)
	Enables the vectorised application of the modified_bessel_first_kind function, when the first and/or second arguments are containers. More...
template<typename T2 , require_arithmetic_t< T2 > * = nullptr>
T2	modified_bessel_second_kind (int v, const T2 z)
	\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \] More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	modified_bessel_second_kind (const T1 &a, const T2 &b)
	Enables the vectorised application of the modified_bessel_second_kind function, when the first and/or second arguments are containers. More...
int	modulus (int x, int y)
template<typename Mat , typename Scal , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>
auto	multiply (const Mat &m, Scal c)
	Return specified matrix multiplied by specified scalar. More...
template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>
auto	multiply (Scal c, const Mat &m)
	Return specified scalar multiplied by specified matrix. More...
template<typename Mat1 , typename Mat2 , require_all_eigen_vt< std::is_arithmetic, Mat1, Mat2 > * = nullptr, require_not_eigen_row_and_col_t< Mat1, Mat2 > * = nullptr>
auto	multiply (const Mat1 &m1, const Mat2 &m2)
	Return the product of the specified matrices. More...
template<typename RowVec , typename ColVec , require_not_var_t< return_type_t< RowVec, ColVec >> * = nullptr, require_eigen_row_and_col_t< RowVec, ColVec > * = nullptr>
auto	multiply (const RowVec &rv, const ColVec &v)
	Return the scalar product of the specified row vector and specified column vector. More...
template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>
return_type_t< Scalar1, Scalar2 >	multiply (Scalar1 m, Scalar2 c)
	Return product of scalars. More...
template<typename T_a , typename T_b , require_all_arithmetic_t< T_a, T_b > * = nullptr>
return_type_t< T_a, T_b >	multiply_log (const T_a a, const T_b b)
	Calculate the value of the first argument times log of the second argument while behaving properly with 0 inputs. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>
auto	multiply_log (const T1 &a, const T2 &b)
	Enables the vectorised application of the multiply_log function, when the first and/or second arguments are containers. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_autodiff< EigMat > * = nullptr>
matrix_d	multiply_lower_tri_self_transpose (const EigMat &L)
	Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
int	num_elements (const T &x)
	Returns 1, the number of elements in a primitive type. More...
template<typename T >
int	num_elements (const std::vector< T > &v)
	Returns the number of elements in the specified vector. More...
template<typename T , typename M , typename S , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr>
auto	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<typename T , typename M , typename S , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr>
auto	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma, return_type_t< T, M, S > &lp)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier, incrementing the specified reference with the log absolute Jacobian determinant of the transform. More...
template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const M &mu, const S &sigma)
	Overload for array of x and non-array mu and sigma. More...
template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const M &mu, const S &sigma, return_type_t< T, M, S > &lp)
	Overload for array of x and non-array mu and sigma with lp. More...
template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const M &mu, const std::vector< S > &sigma)
	Overload for array of x and sigma and non-array mu. More...
template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const M &mu, const std::vector< S > &sigma, return_type_t< T, M, S > &lp)
	Overload for array of x and sigma and non-array mu with lp. More...
template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const std::vector< M > &mu, const S &sigma)
	Overload for array of x and mu and non-array sigma. More...
template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>
auto	offset_multiplier_constrain (const std::vector< T > &x, const std::vector< M > &mu, const S &sigma, return_type_t< T, M, S > &lp)
	Overload for array of x and mu and non-array sigma with lp. More...
template<typename T , typename M , typename S >
auto	offset_multiplier_constrain (const std::vector< T > &x, const std::vector< M > &mu, const std::vector< S > &sigma)
	Overload for array of x, mu, and sigma. More...
template<typename T , typename M , typename S >
auto	offset_multiplier_constrain (const std::vector< T > &x, const std::vector< M > &mu, const std::vector< S > &sigma, return_type_t< T, M, S > &lp)
	Overload for array of x, mu, and sigma with lp. More...
template<typename T , typename M , typename S >
auto	offset_multiplier_free (const T &y, const M &mu, const S &sigma)
	Return the unconstrained variable that transforms to the specified offset and multiplier constrained variable given the specified offset and multiplier. More...
template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>
auto	offset_multiplier_free (const std::vector< T > &x, const M &mu, const S &sigma)
	Overload for array of x and non-array mu and sigma. More...
template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>
auto	offset_multiplier_free (const std::vector< T > &x, const M &mu, const std::vector< S > &sigma)
	Overload for array of x and sigma and non-array mu. More...
template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>
auto	offset_multiplier_free (const std::vector< T > &x, const std::vector< M > &mu, const S &sigma)
	Overload for array of x and mu and non-array sigma. More...
template<typename T , typename M , typename S >
auto	offset_multiplier_free (const std::vector< T > &x, const std::vector< M > &mu, const std::vector< S > &sigma)
	Overload for array of x, mu, and sigma. More...
std::vector< double >	one_hot_array (int K, int k)
	Return an array with 1 in the k-th position and zero elsewhere. More...
std::vector< int >	one_hot_int_array (int K, int k)
	Return an integer array with 1 in the k-th position and zero elsewhere. More...
Eigen::RowVectorXd	one_hot_row_vector (int K, int k)
	Return a row vector with 1 in the k-th position and zero elsewhere. More...
Eigen::VectorXd	one_hot_vector (int K, int k)
	Return a vector with 1 in the k-th position and zero elsewhere. More...
std::vector< double >	ones_array (int K)
	Return an array of ones. More...
std::vector< int >	ones_int_array (int K)
	Return an integer array of ones. More...
auto	ones_row_vector (int K)
	Return a row vector of ones. More...
auto	ones_vector (int K)
	Return a vector of ones. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr, require_not_st_var< EigVec > * = nullptr>
plain_type_t< EigVec >	ordered_constrain (const EigVec &x)
	Return an increasing ordered vector derived from the specified free vector. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>
auto	ordered_constrain (const EigVec &x, value_type_t< EigVec > &lp)
	Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	ordered_constrain (const T &x, return_type_t< T > &lp)
	Return a positive valued, increasing ordered vector derived from the specified free vector. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>
plain_type_t< EigVec >	ordered_free (const EigVec &y)
	Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	ordered_free (const T &x)
	Overload of ordered_free() to untransform each Eigen vector in a standard vector. More...
double	owens_t (double h, double a)
	Return the result of applying Owen's T function to the specified arguments. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	owens_t (const T1 &a, const T2 &b)
	Enables the vectorised application of the owens_t function, when the first and/or second arguments are containers. More...
double	Phi (double x)
	The unit normal cumulative distribution function. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	Phi (const T &x)
	Vectorized version of Phi(). More...
double	Phi_approx (double x)
	Return an approximation of the unit normal CDF. More...
double	Phi_approx (int x)
	Return an approximation of the unit normal CDF. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	Phi_approx (const T &x)
	Return the elementwise application of Phi_approx() to specified argument container. More...
template<typename T >
T	plus (T &&x)
	Returns the unary plus of the input. More...
template<typename T_theta , typename T_scalar = scalar_type_t<T_theta>, require_eigen_vector_t< T_theta > * = nullptr>
plain_type_t< T_theta >	poisson_binomial_log_probs (int y, const T_theta &theta)
	Returns the last row of the log probability matrix of the Poisson-Binomial distribution given the number of successes and a vector of success probabilities. More...
template<typename T_y , typename T_theta , require_vt_integral< T_y > * = nullptr>
auto	poisson_binomial_log_probs (const T_y &y, const T_theta &theta)
std::complex< double >	polar (double r, double theta)
	Returns the complex number with specified magnitude and phase angle. More...
std::complex< double >	polar (double r, int theta)
std::complex< double >	polar (int r, double theta)
std::complex< double >	polar (int r, int theta)
template<typename T >
auto	positive_constrain (const T &x)
	Return the positive value for the specified unconstrained input. More...
template<typename T , typename S >
auto	positive_constrain (const T &x, S &lp)
	Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	positive_constrain (const T &x, return_type_t< T > &lp)
	Return the positive value for the specified unconstrained input. More...
template<typename T >
T	positive_free (const T &y)
	Return the unconstrained value corresponding to the specified positive-constrained value. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr, require_not_st_var< EigVec > * = nullptr>
auto	positive_ordered_constrain (const EigVec &x)
	Return an increasing positive ordered vector derived from the specified free vector. More...
template<typename Vec , require_col_vector_t< Vec > * = nullptr>
auto	positive_ordered_constrain (const Vec &x, return_type_t< Vec > &lp)
	Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<bool Jacobian, typename Vec , require_not_std_vector_t< Vec > * = nullptr>
auto	positive_ordered_constrain (const Vec &x, return_type_t< Vec > &lp)
	Return a positive valued, increasing positive ordered vector derived from the specified free vector. More...
template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>
auto	positive_ordered_constrain (const T &x, return_type_t< T > &lp)
	Return a positive valued, increasing positive ordered vector derived from the specified free vector. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>
auto	positive_ordered_free (const EigVec &y)
	Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	positive_ordered_free (const T &x)
	Overload of positive_ordered_free() to untransform each Eigen vector in a standard vector. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	pow (const T1 &a, const T2 &b)
	Enables the vectorised application of the pow function, when the first and/or second arguments are containers. More...
template<typename T >
std::enable_if< std::is_arithmetic< T >::value, T >::type	primitive_value (T x)
	Return the value of the specified arithmetic argument unmodified with its own declared type. More...
template<typename T >
std::enable_if<!std::is_arithmetic< T >::value, double >::type	primitive_value (const T &x)
	Return the primitive value of the specified argument. More...
template<typename T >
T	prob_constrain (const T &x)
	Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar. More...
template<typename T >
T	prob_constrain (const T &x, T &lp)
	Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<bool Jacobian, typename T >
auto	prob_constrain (const T &x, T &lp)
	Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar. More...
template<typename T >
T	prob_free (const T &y)
	Return the free scalar that when transformed to a probability produces the specified scalar. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
T	prod (const T &v)
	Returns the product of given scalar. More...
template<typename T >
T	prod (const std::vector< T > &v)
	Returns the product of the coefficients of the specified standard vector. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
value_type_t< EigMat >	prod (const EigMat &v)
	Returns the product of the coefficients of the specified column vector. More...
template<typename T , typename S >
promote_scalar_type< T, S >::type	promote_scalar (const S &x)
	This is the top-level function to call to promote the scalar types of an input of type S to type T. More...
template<typename T >
Eigen::Matrix< T,-1,-1 >	pseudo_eigenvalues (const Eigen::Matrix< T,-1,-1 > &m)
template<typename T >
Eigen::Matrix< T,-1,-1 >	pseudo_eigenvectors (const Eigen::Matrix< T,-1,-1 > &m)
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	qr_Q (const EigMat &m)
	Returns the orthogonal factor of the fat QR decomposition. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	qr_R (const EigMat &m)
	Returns the upper triangular factor of the fat QR decomposition. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	qr_thin_Q (const EigMat &m)
	Returns the orthogonal factor of the thin QR decomposition. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	qr_thin_R (const EigMat &m)
	Returns the upper triangular factor of the thin QR decomposition. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr, require_all_vt_arithmetic< EigMat1, EigMat2 > * = nullptr>
auto	quad_form (const EigMat1 &A, const EigMat2 &B)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_vt_same< EigMat, ColVec > * = nullptr, require_all_vt_arithmetic< EigMat, ColVec > * = nullptr>
value_type_t< EigMat >	quad_form (const EigMat &A, const ColVec &B)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat , typename EigVec , require_eigen_t< EigMat > * = nullptr, require_eigen_vector_t< EigVec > * = nullptr>
auto	quad_form_diag (const EigMat &mat, const EigVec &vec)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr, require_all_vt_arithmetic< EigMat1, EigMat2 > * = nullptr>
plain_type_t< EigMat2 >	quad_form_sym (const EigMat1 &A, const EigMat2 &B)
	Return the quadratic form \( B^T A B \) of a symmetric matrix. More...
template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_vt_same< EigMat, ColVec > * = nullptr, require_all_vt_arithmetic< EigMat, ColVec > * = nullptr>
value_type_t< EigMat >	quad_form_sym (const EigMat &A, const ColVec &B)
	Return the quadratic form \( B^T A B \) of a symmetric matrix. More...
template<typename T , require_vector_t< T > * = nullptr, require_vector_vt< std::is_arithmetic, T > * = nullptr>
double	quantile (const T &samples_vec, const double p)
	Return sample quantiles corresponding to the given probabilities. More...
template<typename T , typename Tp , require_all_vector_t< T, Tp > * = nullptr, require_vector_vt< std::is_arithmetic, T > * = nullptr, require_std_vector_vt< std::is_arithmetic, Tp > * = nullptr>
std::vector< double >	quantile (const T &samples_vec, const Tp &ps)
	Return sample quantiles corresponding to the given probabilities. More...
template<typename C , require_container_t< C > * = nullptr>
int	rank (const C &v, int s)
	Return the number of components of v less than v[s]. More...
template<typename T , require_eigen_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	read_corr_L (const T &CPCs, size_t K)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T , require_eigen_vector_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	read_corr_L (const T &CPCs, size_t K, value_type_t< T > &log_prob)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T_CPCs , require_eigen_vector_t< T_CPCs > * = nullptr>
Eigen::Matrix< value_type_t< T_CPCs >, Eigen::Dynamic, Eigen::Dynamic >	read_corr_matrix (const T_CPCs &CPCs, size_t K)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T_CPCs , require_eigen_vector_t< T_CPCs > * = nullptr>
Eigen::Matrix< value_type_t< T_CPCs >, Eigen::Dynamic, Eigen::Dynamic >	read_corr_matrix (const T_CPCs &CPCs, size_t K, value_type_t< T_CPCs > &log_prob)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>
Eigen::Matrix< value_type_t< T_CPCs >, Eigen::Dynamic, Eigen::Dynamic >	read_cov_L (const T_CPCs &CPCs, const T_sds &sds, value_type_t< T_CPCs > &log_prob)
	This is the function that should be called prior to evaluating the density of any elliptical distribution. More...
template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>
Eigen::Matrix< value_type_t< T_CPCs >, Eigen::Dynamic, Eigen::Dynamic >	read_cov_matrix (const T_CPCs &CPCs, const T_sds &sds, value_type_t< T_CPCs > &log_prob)
	A generally worse alternative to call prior to evaluating the density of an elliptical distribution. More...
template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>
Eigen::Matrix< value_type_t< T_CPCs >, Eigen::Dynamic, Eigen::Dynamic >	read_cov_matrix (const T_CPCs &CPCs, const T_sds &sds)
	Builds a covariance matrix from CPCs and standard deviations. More...
template<typename T , require_autodiff_t< T > >
T	real (const std::complex< T > &z)
	Return the real component of the complex argument. More...
template<typename T_ret , typename In , require_std_vector_t< T_ret > * = nullptr>
std::vector< plain_type_t< In > >	rep_array (const In &x, int n)
template<typename In >
std::vector< plain_type_t< In > >	rep_array (const In &x, int n)
template<typename In >
std::vector< std::vector< plain_type_t< In > > >	rep_array (const In &x, int m, int n)
template<typename In >
std::vector< std::vector< std::vector< plain_type_t< In > > > >	rep_array (const In &x, int k, int m, int n)
template<typename Ret , typename T , require_eigen_matrix_dynamic_vt< is_stan_scalar, Ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>
auto	rep_matrix (const T &x, int m, int n)
	Implementation of rep_matrix returning an Eigen matrix with scalar type equal to the input scalar type. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
auto	rep_matrix (const T &x, int m, int n)
	Default Implementation of rep_matrix returning an Eigen matrix with scalar type equal to the input scalar type. More...
template<typename Vec , require_eigen_vector_t< Vec > * = nullptr>
auto	rep_matrix (const Vec &x, int n)
	Implementation of rep_matrix returning an Eigen matrix from an Eigen vector. More...
template<typename T_ret , typename T , require_eigen_row_vector_t< T_ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>
auto	rep_row_vector (const T &x, int n)
template<typename T , require_stan_scalar_t< T > * = nullptr>
auto	rep_row_vector (const T &x, int n)
template<typename T_ret , typename T , require_eigen_col_vector_t< T_ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>
auto	rep_vector (const T &x, int n)
template<typename T , require_stan_scalar_t< T > * = nullptr>
auto	rep_vector (const T &x, int n)
template<typename T >
void	resize (T &x, std::vector< int > dims)
	Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices. More...
template<typename T >
std::vector< T >	reverse (const std::vector< T > &x)
	Return a copy of the specified array in reversed order. More...
template<typename T , typename = require_vector_t<T>>
auto	reverse (const T &x)
	Return a copy of the specified vector or row vector in reversed order. More...
template<typename T , require_arithmetic_t< T > * = nullptr>
return_type_t< T >	rising_factorial (const T &x, int n)
	Return the rising factorial function evaluated at the inputs. More...
template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>
auto	rising_factorial (const T1 &a, const T2 &b)
	Enables the vectorised application of the rising_factorial function, when the first and/or second arguments are containers. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	round (const Container &x)
	Vectorized version of round(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	round (const Container &x)
	Version of round() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	row (const T &m, size_t i)
	Return the specified row of the specified matrix, using start-at-1 indexing. More...
template<typename T , require_matrix_t< T > * = nullptr>
int	rows (const T &m)
	Return the number of rows in the specified matrix, vector, or row vector. More...
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>
Eigen::Matrix< return_type_t< Mat1, Mat2 >, Mat1::RowsAtCompileTime, 1 >	rows_dot_product (const Mat1 &v1, const Mat2 &v2)
	Returns the dot product of rows of the specified matrices. More...
template<typename T , require_eigen_t< T > * = nullptr, require_not_st_var< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, T::RowsAtCompileTime, 1 >	rows_dot_self (const T &x)
	Returns the dot product of each row of a matrix with itself. More...
template<typename T , typename = require_autodiff_t<T>>
double	scalbn (const T &x, int n)
template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< std::is_arithmetic, EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< double, Eigen::Dynamic, EigMat2::ColsAtCompileTime >	scale_matrix_exp_multiply (const double &t, const EigMat1 &A, const EigMat2 &B)
	Return product of exp(At) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double. More...
template<typename Tt , typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_autodiff_t< Tt, value_type_t< EigMat1 >, value_type_t< EigMat2 >> * = nullptr>
Eigen::Matrix< return_type_t< Tt, EigMat1, EigMat2 >, Eigen::Dynamic, EigMat2::ColsAtCompileTime >	scale_matrix_exp_multiply (const Tt &t, const EigMat1 &A, const EigMat2 &B)
	Return product of exp(At) and B, where A is a NxN matrix, B is a NxCb matrix and t is a scalar. More...
void	scaled_add (std::vector< double > &x, const std::vector< double > &y, double lambda)
template<typename T , require_container_t< T > * = nullptr, require_not_st_var< T > * = nullptr>
auto	sd (const T &m)
	Returns the unbiased sample standard deviation of the coefficients in the specified std vector, column vector, row vector, or matrix. More...
template<typename Vec , require_vector_t< Vec > * = nullptr>
auto	segment (const Vec &v, size_t i, size_t n)
	Return the specified number of elements as a row/column vector starting from the specified element - 1 of the specified row/column vector. More...
template<typename T >
std::vector< T >	segment (const std::vector< T > &sv, size_t i, size_t n)
template<typename T , require_stan_scalar_t< T > * = nullptr>
int	sign (const T &z)
template<typename ADType , require_autodiff_t< ADType > * = nullptr>
bool	signbit (ADType &&v)
	Return true if the specified argument is negative and false otherwise. More...
template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr, require_not_st_var< Vec > * = nullptr>
auto	simplex_constrain (const Vec &y)
	Return the simplex corresponding to the specified free vector. More...
template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr, require_not_st_var< Vec > * = nullptr>
auto	simplex_constrain (const Vec &y, value_type_t< Vec > &lp)
	Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<bool Jacobian, typename Vec , require_not_std_vector_t< Vec > * = nullptr>
auto	simplex_constrain (const Vec &y, return_type_t< Vec > &lp)
	Return the simplex corresponding to the specified free vector. More...
template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>
auto	simplex_constrain (const T &y, return_type_t< T > &lp)
	Return the simplex corresponding to the specified free vector. More...
template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr>
auto	simplex_free (const Vec &x)
	Return an unconstrained vector that when transformed produces the specified simplex. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	simplex_free (const T &x)
	Overload of simplex_free() to untransform each Eigen vector in a standard vector. More...
template<typename T , require_not_container_st< std::is_arithmetic, T > * = nullptr, require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	sin (const T &x)
	Vectorized version of sin(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	sin (const Container &x)
	Version of sin() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, 1 >	singular_values (const EigMat &m)
	Return the vector of the singular values of the specified matrix in decreasing order of magnitude. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	sinh (const Container &x)
	Vectorized version of sinh(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	sinh (const Container &x)
	Version of sinh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename ScalarT , require_stan_scalar_t< ScalarT > * = nullptr>
size_t	size_mvt (const ScalarT &)
	Provides the size of a multivariate argument. More...
template<typename MatrixT , require_matrix_t< MatrixT > * = nullptr>
size_t	size_mvt (const MatrixT &)
template<typename MatrixT , require_matrix_t< MatrixT > * = nullptr>
size_t	size_mvt (const std::vector< MatrixT > &x)
template<typename T >
bool	size_zero (const T &x)
	Returns 1 if input is of length 0, returns 0 otherwise. More...
template<typename T , typename... Ts>
bool	size_zero (const T &x, const Ts &...xs)
	Returns 1 if any inputs are of length 0, returns 0 otherwise. More...
template<typename ColVec , require_eigen_col_vector_vt< std::is_arithmetic, ColVec > * = nullptr>
plain_type_t< ColVec >	softmax (const ColVec &v)
	Return the softmax of the specified vector. More...
template<typename T >
std::vector< T >	sort_asc (std::vector< T > xs)
	Return the specified standard vector in ascending order. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
plain_type_t< EigVec >	sort_asc (EigVec &&xs)
	Return the specified vector in ascending order. More...
template<typename T >
std::vector< T >	sort_desc (std::vector< T > xs)
	Return the specified standard vector in descending order. More...
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
plain_type_t< EigVec >	sort_desc (EigVec &&xs)
	Return the specified vector in descending order. More...
template<bool ascending, typename C >
std::vector< int >	sort_indices (const C &xs)
	Return an integer array of indices of the specified container sorting the values in ascending or descending order based on the value of the first template parameter. More...
template<typename C >
std::vector< int >	sort_indices_asc (const C &xs)
	Return a sorted copy of the argument container in ascending order. More...
template<typename C >
std::vector< int >	sort_indices_desc (const C &xs)
	Return a sorted copy of the argument container in ascending order. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>
auto	sqrt (const Container &x)
	Vectorized version of sqrt(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>
auto	sqrt (const Container &x)
	Version of sqrt() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
double	square (double x)
	Return the square of the specified argument. More...
template<typename Container , require_not_stan_scalar_t< Container > * = nullptr, require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	square (const Container &x)
	Vectorized version of square(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	square (const Container &x)
	Version of square() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr, require_all_not_var_t< Scal1, Scal2 > * = nullptr>
return_type_t< Scal1, Scal2 >	squared_distance (const Scal1 &x1, const Scal2 &x2)
	Returns the squared distance. More...
template<typename EigVec1 , typename EigVec2 , require_all_eigen_vector_t< EigVec1, EigVec2 > * = nullptr, require_all_not_eigen_vt< is_var, EigVec1, EigVec2 > * = nullptr>
return_type_t< EigVec1, EigVec2 >	squared_distance (const EigVec1 &v1, const EigVec2 &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , require_not_container_t< T > * = nullptr>
void	stan_print (std::ostream *o, const T &x)
template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>
void	stan_print (std::ostream *o, const EigVec &x)
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vector_t< EigMat > * = nullptr>
void	stan_print (std::ostream *o, const EigMat &x)
template<typename T >
void	stan_print (std::ostream *o, const std::vector< T > &x)
template<typename T , require_stan_scalar_t< T > * = nullptr>
T	step (const T &y)
	The step, or Heaviside, function. More...
template<typename T , require_container_t< T > * = nullptr>
auto	step (const T &x)
	Return the elementwise application of step() to specified argument container. More...
template<typename T , require_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	sub_col (const T &m, size_t i, size_t j, size_t nrows)
	Return a nrows x 1 subcolumn starting at (i-1, j-1). More...
template<typename T , require_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	sub_row (const T &m, size_t i, size_t j, size_t ncols)
	Return a 1 x ncols subrow starting at (i-1, j-1). More...
template<typename ScalarA , typename ScalarB , require_all_stan_scalar_t< ScalarA, ScalarB > * = nullptr, require_all_not_var_t< ScalarA, ScalarB > * = nullptr>
return_type_t< ScalarA, ScalarB >	subtract (const ScalarA &a, const ScalarB &b)
	Return the result of subtracting the second scalar from the first scalar. More...
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>
auto	subtract (const Mat1 &m1, const Mat2 &m2)
	Return the result of subtracting the second specified matrix from the first specified matrix. More...
template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Mat, Scal > * = nullptr>
auto	subtract (const Scal c, const Mat &m)
	Return the result of subtracting the specified matrix from the specified scalar. More...
template<typename Mat , typename Scal , require_eigen_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>
auto	subtract (const Mat &m, const Scal c)
	Return the result of subtracting the specified scalar from the specified matrix. More...
template<typename T , require_stan_scalar_t< T > * = nullptr>
T	sum (T &&m)
	Returns specified input value. More...
template<typename T , require_not_var_t< T > * = nullptr>
T	sum (const std::vector< T > &m)
	Return the sum of the values in the specified standard vector. More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	svd_U (const EigMat &m)
	Given input matrix m, return matrix U where m = UDV^{T} More...
template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	svd_V (const EigMat &m)
	Given input matrix m, return matrix V where m = UDV^{T} More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	symmetrize_from_lower_tri (const T &m)
	Return a symmetric matrix using elements from the lower triangular part of the input matrix. More...
template<typename T , require_eigen_t< T > * = nullptr>
Eigen::Matrix< value_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	symmetrize_from_upper_tri (const T &m)
	Return a symmetric matrix using elements from the upper triangular part of the input matrix. More...
template<typename T , require_vector_t< T > * = nullptr>
auto	tail (const T &v, size_t n)
	Return the specified number of elements as a vector or row vector (same as input) from the back of the specified vector or row vector. More...
template<typename T >
std::vector< T >	tail (const std::vector< T > &sv, size_t n)
	Return the specified number of elements as a standard vector from the back of the specified standard vector. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	tan (const Container &x)
	Vectorized version of tan(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	tan (const Container &x)
	Version of tan() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>
auto	tanh (const Container &x)
	Vectorized version of tanh(). More...
template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>
auto	tanh (const Container &x)
	Version of tanh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these. More...
template<typename T , require_eigen_vt< std::is_arithmetic, T > * = nullptr>
Eigen::Matrix< value_type_t< T >, T::RowsAtCompileTime, T::RowsAtCompileTime >	tcrossprod (const T &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...
double	tgamma (double x)
	Return the gamma function applied to the specified argument. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
auto	tgamma (const T &x)
	Vectorized version of tgamma(). More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
std::vector< value_type_t< EigMat > >	to_array_1d (const EigMat &matrix)
template<typename T >
std::vector< T >	to_array_1d (const std::vector< T > &x)
template<typename T >
std::vector< typename scalar_type< T >::type >	to_array_1d (const std::vector< std::vector< T >> &x)
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
std::vector< std::vector< value_type_t< EigMat > > >	to_array_2d (const EigMat &matrix)
template<typename T = double, typename S = double>
constexpr std::complex< stan::real_return_t< T, S > >	to_complex (const T &re=0, const S &im=0)
	Return a complex value from a real component and an imaginary component. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
EigMat	to_matrix (EigMat &&x)
	Returns a matrix with dynamic dimensions constructed from an Eigen matrix which is either a row vector, column vector, or matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< Eigen::Matrix< T, 1, Eigen::Dynamic >> &x)
	Returns a matrix representation of a standard vector of Eigen row vectors with the same dimensions and indexing order. More...
template<typename T >
Eigen::Matrix< return_type_t< T, double >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< std::vector< T >> &x)
	Returns a matrix representation of the standard vector of standard vectors with the same dimensions and indexing order. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (EigMat &&x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T >
auto	to_matrix (const std::vector< T > &x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< int > &x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (EigMat &&x, int m, int n, bool col_major)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T >
Eigen::Matrix< return_type_t< T, double >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< T > &x, int m, int n, bool col_major)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T >
ref_type_t< T && >	to_ref (T &&a)
	This evaluates expensive Eigen expressions. More...
template<bool Cond, typename T , std::enable_if_t<!Cond > * = nullptr>
T	to_ref_if (T &&a)
	No-op that should be optimized away. More...
template<bool Cond, typename T , std::enable_if_t< Cond > * = nullptr>
ref_type_t< T && >	to_ref_if (T &&a)
	If the condition is true, evaluates expensive Eigen expressions. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, 1, Eigen::Dynamic >	to_row_vector (const EigMat &matrix)
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	to_row_vector (const std::vector< T > &vec)
Eigen::Matrix< double, 1, Eigen::Dynamic >	to_row_vector (const std::vector< int > &vec)
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
Eigen::Matrix< value_type_t< EigMat >, Eigen::Dynamic, 1 >	to_vector (const EigMat &matrix)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	to_vector (const std::vector< T > &vec)
Eigen::Matrix< double, Eigen::Dynamic, 1 >	to_vector (const std::vector< int > &vec)
template<typename T , require_eigen_t< T > * = nullptr, require_not_st_var< T > * = nullptr>
value_type_t< T >	trace (const T &m)
	Returns the trace of the specified matrix. More...
template<typename EigMat1 , typename T2 , typename EigMat3 , require_not_col_vector_t< EigMat1 > * = nullptr, require_all_not_st_var< EigMat1, T2, EigMat3 > * = nullptr>
return_type_t< EigMat1, T2, EigMat3 >	trace_gen_inv_quad_form_ldlt (const EigMat1 &D, LDLT_factor< T2 > &A, const EigMat3 &B)
	Compute the trace of an inverse quadratic form. More...
template<typename EigVec , typename T , typename EigMat , require_col_vector_t< EigVec > * = nullptr, require_all_not_st_var< EigVec, T, EigMat > * = nullptr>
return_type_t< EigVec, T, EigMat >	trace_gen_inv_quad_form_ldlt (const EigVec &D, LDLT_factor< T > &A, const EigMat &B)
	Compute the trace of an inverse quadratic form. More...
template<typename TD , typename TA , typename TB , typename = require_all_eigen_t<TD, TA, TB>, typename = require_all_not_vt_var<TD, TA, TB>, typename = require_any_not_vt_arithmetic<TD, TA, TB>>
auto	trace_gen_quad_form (const TD &D, const TA &A, const TB &B)
	Return the trace of D times the quadratic form of B and A. More...
template<typename EigMatD , typename EigMatA , typename EigMatB , require_all_eigen_vt< std::is_arithmetic, EigMatD, EigMatA, EigMatB > * = nullptr>
double	trace_gen_quad_form (const EigMatD &D, const EigMatA &A, const EigMatB &B)
	Return the trace of D times the quadratic form of B and A. More...
template<typename T , typename EigMat2 , typename = require_all_not_st_var<T, EigMat2>>
return_type_t< T, EigMat2 >	trace_inv_quad_form_ldlt (LDLT_factor< T > &A, const EigMat2 &B)
	Compute the trace of an inverse quadratic form. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< std::is_arithmetic, EigMat1, EigMat2 > * = nullptr>
return_type_t< EigMat1, EigMat2 >	trace_quad_form (const EigMat1 &A, const EigMat2 &B)
	Compute trace(B^T A B). More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	transpose (const T &m)
	Transposes a matrix. More...
template<typename T >
T	trigamma_impl (const T &x)
	Return the trigamma function applied to the argument. More...
double	trigamma (double u)
	Return the second derivative of the log Gamma function evaluated at the specified argument. More...
double	trigamma (int u)
	Return the second derivative of the log Gamma function evaluated at the specified argument. More...
template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	trigamma (const T &x)
	Return the elementwise application of trigamma() to specified argument container. More...
template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>
auto	trunc (const T &x)
	Return the elementwise application of trunc() to specified argument container. More...
template<typename T , typename U , require_all_stan_scalar_t< T, U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub)
	Return the upper-bounded value for the specified unconstrained matrix and upper bound. More...
template<typename T , typename U , require_all_stan_scalar_t< T, U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub, std::decay_t< return_type_t< T, U >> &lp)
	Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T , typename U , require_eigen_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub)
	Specialization of ub_constrain to apply a scalar upper bound elementwise to each input. More...
template<typename T , typename U , require_eigen_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub, std::decay_t< return_type_t< T, U >> &lp)
	Specialization of ub_constrain to apply a scalar upper bound elementwise to each input. More...
template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>
auto	ub_constrain (const std::vector< T > &x, const U &ub)
	Specialization of ub_constrain to apply a scalar upper bound elementwise to each input element. More...
template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>
auto	ub_constrain (const std::vector< T > &x, const U &ub, return_type_t< T, U > &lp)
	Specialization of ub_constrain to apply a scalar upper bound elementwise to each input element. More...
template<typename T , typename U >
auto	ub_constrain (const std::vector< T > &x, const std::vector< U > &ub)
	Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element. More...
template<typename T , typename U >
auto	ub_constrain (const std::vector< T > &x, const std::vector< U > &ub, return_type_t< T, U > &lp)
	Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element. More...
template<bool Jacobian, typename T , typename U >
auto	ub_constrain (const T &x, const U &ub, return_type_t< T, U > &lp)
	Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element. More...
template<typename T , typename U , require_not_std_vector_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr>
auto	ub_free (T &&y, U &&ub)
	Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound. More...
template<typename T , typename U , require_all_eigen_t< T, U > * = nullptr>
auto	ub_free (T &&y, U &&ub)
	Return the free matrix that corresponds to the specified upper-bounded matrix with respect to the specified upper bound. More...
template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>
auto	ub_free (const std::vector< T > y, const U &ub)
	Return the free variable that corresponds to the specified upper-bounded variable with respect to the specified upper bound. More...
template<typename T , typename U >
auto	ub_free (const std::vector< T > y, const std::vector< U > &ub)
	Return the free variable that corresponds to the specified upper-bounded variable with respect to the specified upper bound. More...
auto	uniform_simplex (int K)
	Return a uniform simplex of size K. More...
template<typename T , require_eigen_col_vector_t< T > * = nullptr, require_not_vt_autodiff< T > * = nullptr>
plain_type_t< T >	unit_vector_constrain (const T &y)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T1 , typename T2 , require_eigen_col_vector_t< T1 > * = nullptr, require_all_not_vt_autodiff< T1, T2 > * = nullptr>
plain_type_t< T1 >	unit_vector_constrain (const T1 &y, T2 &lp)
	Return the unit length vector corresponding to the free vector y. More...
template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>
auto	unit_vector_constrain (const T &y, return_type_t< T > &lp)
	Return the unit length vector corresponding to the free vector y. More...
template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>
auto	unit_vector_free (EigVec &&x)
	Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1. More...
template<typename T , require_std_vector_t< T > * = nullptr>
auto	unit_vector_free (const T &x)
	Overload of unit_vector_free() to untransform each Eigen vector in a standard vector. More...
std::vector< int >	unitspaced_array (int low, int high)
	Return an array of integers in an ordered sequence. More...
template<typename T , require_st_arithmetic< T > * = nullptr>
T	value_of (T &&x)
	Inputs that are arithmetic types or containers of airthmetric types are returned from value_of unchanged. More...
template<typename T , require_complex_t< T > * = nullptr, require_t< std::is_arithmetic< typename std::decay_t< T >::value_type >> * = nullptr>
auto	value_of (T &&x)
template<typename T , require_std_vector_t< T > * = nullptr, require_not_st_arithmetic< T > * = nullptr>
auto	value_of (const T &x)
	For std::vectors of non-arithmetic types, return a std::vector composed of value_of applied to each element. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_st_arithmetic< EigMat > * = nullptr>
auto	value_of (EigMat &&M)
	For Eigen matrices and expressions of non-arithmetic types, return an expression that represents the Eigen::Matrix resulting from applying value_of elementwise. More...
template<typename T , typename = require_stan_scalar_t<T>>
double	value_of_rec (const T x)
	Return the value of the specified scalar argument converted to a double value. More...
double	value_of_rec (double x)
	Return the specified argument. More...
template<typename T >
std::complex< double >	value_of_rec (const std::complex< T > &x)
	Recursively apply value-of to the parts of the argument. More...
template<typename T , require_not_same_t< double, T > * = nullptr>
std::vector< double >	value_of_rec (const std::vector< T > &x)
	Convert a std::vector of type T to a std::vector of doubles. More...
template<typename T , require_std_vector_t< T > * = nullptr, require_vt_same< double, T > * = nullptr>
T	value_of_rec (T &&x)
	Return the specified argument. More...
template<typename T , typename = require_not_st_same<T, double>, typename = require_eigen_t<T>>
auto	value_of_rec (T &&M)
	Convert a matrix of type T to a matrix of doubles. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>
value_type_t< EigMat >	variance (const EigMat &m)
	Returns the sample variance (divide by length - 1) of the coefficients in the specified matrix. More...
template<typename StdVec , require_std_vector_t< StdVec > * = nullptr, require_not_vt_var< StdVec > * = nullptr>
value_type_t< StdVec >	variance (const StdVec &v)
	Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector. More...
template<typename Vec , typename... Args>
auto	vec_concat (const Vec &v1, const Args &...args)
	Get the event stack from a vector of events and other arguments. More...
std::vector< double >	zeros_array (int K)
	Return an array of zeros. More...
std::vector< int >	zeros_int_array (int K)
	Return an integer array of zeros. More...
auto	zeros_row_vector (int K)
	Return a row vector of zeros. More...
auto	zeros_vector (int K)
	Return a vector of zeros. More...
template<class F , class Tuple >
decltype(auto) constexpr	apply (F &&f, Tuple &&t)
template<typename T1 , typename T2 , typename F , require_all_stan_scalar_t< T1, T2 > * = nullptr>
auto	apply_scalar_binary (const T1 &x, const T2 &y, const F &f)
	Base template function for vectorization of binary scalar functions defined by applying a functor to a combination of scalars, containers of matching sizes, or a combination of a scalar and a container. More...
template<typename T1 , typename T2 , typename F , require_eigen_vector_vt< is_stan_scalar, T1 > * = nullptr, require_std_vector_vt< std::is_integral, T2 > * = nullptr>
auto	apply_scalar_binary (const T1 &x, const T2 &y, const F &f)
	Specialisation for use with one Eigen vector (row or column) and a one-dimensional std::vector of integer types. More...
template<typename T1 , typename T2 , typename F , require_eigen_matrix_dynamic_vt< is_stan_scalar, T1 > * = nullptr, require_std_vector_vt< is_std_vector, T2 > * = nullptr, require_std_vector_st< std::is_integral, T2 > * = nullptr>
auto	apply_scalar_binary (const T1 &x, const T2 &y, const F &f)
	Specialisation for use with one Eigen matrix and a two-dimensional std::vector of integer types. More...
template<typename F >
void	finite_diff_gradient (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx, double epsilon=1e-03)
	Calculate the value and the gradient of the specified function at the specified argument using finite difference. More...
template<typename F >
void	finite_diff_gradient_auto (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument using finite difference. More...
template<typename F , typename T >
constexpr auto	for_each (F &&f, T &&t)
	Apply a function to each element of a tuple. More...
template<typename F , typename T1 , typename T2 >
constexpr auto	for_each (F &&f, T1 &&t1, T2 &&t2)
	Apply a function to each element of two tuples. More...
template<typename F , typename T1 , typename T2 , typename T3 >
constexpr auto	for_each (F &&f, T1 &&t1, T2 &&t2, T3 &&t3)
	Apply a function to each element of three tuples. More...
template<typename F >
double	integrate (const F &f, double a, double b, double relative_tolerance)
	Integrate a single variable function f from a to b to within a specified relative tolerance. More...
template<typename F , typename... Args, require_all_not_st_var< Args... > * = nullptr>
double	integrate_1d_impl (const F &f, double a, double b, double relative_tolerance, std::ostream *msgs, const Args &...args)
	Compute the integral of the single variable function f from a to b to within a specified relative tolerance. More...
template<typename F >
double	integrate_1d (const F &f, double a, double b, const std::vector< double > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, std::ostream *msgs, const double relative_tolerance=std::sqrt(EPSILON))
	Compute the integral of the single variable function f from a to b to within a specified relative tolerance. More...
template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >
auto	integrate_ode_rk45 (const F &f, const std::vector< T_y0 > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T_param > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-6, double absolute_tolerance=1e-6, int max_num_steps=1e6)
template<int call_id, typename F , typename T_shared_param , typename T_job_param , require_eigen_col_vector_t< T_shared_param > * = nullptr>
Eigen::Matrix< return_type_t< T_shared_param, T_job_param >, Eigen::Dynamic, 1 >	map_rect (const T_shared_param &shared_params, const std::vector< Eigen::Matrix< T_job_param, Eigen::Dynamic, 1 >> &job_params, const std::vector< std::vector< double >> &x_r, const std::vector< std::vector< int >> &x_i, std::ostream *msgs=nullptr)
	Map N function evaluations to parameters and data which are in rectangular format. More...
std::vector< int >	mpi_map_chunks (std::size_t num_jobs, std::size_t chunk_size=1)
	Maps jobs of given chunk size to workers and returning a vector of counts. More...
template<typename T >
std::unique_lock< std::mutex >	mpi_broadcast_command ()
	Broadcasts default constructible commands to the cluster. More...
std::unique_lock< std::mutex >	mpi_broadcast_command (std::shared_ptr< mpi_command > &command)
	Broadcasts a command instance to the listening cluster. More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_ckrk_tol_impl (const char *function_name, const F &f, const T_y0 &y0_arg, T_t0 t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_ckrk_tol (const F &f, const T_y0 &y0_arg, T_t0 t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_ckrk (const F &f, const T_y0 &y0, T_t0 t0, const std::vector< T_ts > &ts, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_rk45_tol_impl (const char *function_name, const F &f, const T_y0 &y0_arg, T_t0 t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_rk45_tol (const F &f, const T_y0 &y0_arg, T_t0 t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, Args... >, Eigen::Dynamic, 1 > >	ode_rk45 (const F &f, const T_y0 &y0, T_t0 t0, const std::vector< T_ts > &ts, std::ostream *msgs, const Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0_t0 , typename T_t0 , typename T_t , typename... Args, typename = require_all_arithmetic_t<T_y0_t0, T_t0, T_t, scalar_type_t<Args>...>>
Eigen::VectorXd	ode_store_sensitivities (const F &f, const std::vector< double > &coupled_state, const Eigen::Matrix< T_y0_t0, Eigen::Dynamic, 1 > &y0, T_t0 t0, T_t t, std::ostream *msgs, const Args &...args)
	When all arguments are arithmetic, there are no sensitivities to store, so the function just returns the current coupled_state. More...
template<typename ReduceFunction , typename Vec , typename = require_vector_like_t<Vec>, typename... Args>
auto	reduce_sum (Vec &&vmapped, int grainsize, std::ostream *msgs, Args &&...args)
	Call an instance of the function ReduceFunction on every element of an input sequence and sum these terms. More...
template<typename ReduceFunction , typename Vec , typename = require_vector_like_t<Vec>, typename... Args>
auto	reduce_sum_static (Vec &&vmapped, int grainsize, std::ostream *msgs, Args &&...args)
	Call an instance of the function ReduceFunction on every element of an input sequence and sum these terms. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t<std::is_same<std::decay_t<T_actual>, std::decay_t<T_desired>>::value && !is_eigen<T_desired>::value>>
T_actual &&	forward_as (T_actual &&a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t< !std::is_same<std::decay<T_actual>, std::decay<T_desired>>::value && (!is_eigen<T_desired>::value || !is_eigen<T_actual>::value)>>
T_desired	forward_as (const T_actual &a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , require_eigen_t< T_desired > * = nullptr, std::enable_if_t< std::is_same< value_type_t< T_actual >, value_type_t< T_desired >>::value &&is_eigen< T_desired >::value &&is_eigen< T_actual >::value &&internal::eigen_static_size_match(T_desired::RowsAtCompileTime, std::decay_t< T_actual >::RowsAtCompileTime)&&internal::eigen_static_size_match(T_desired::ColsAtCompileTime, std::decay_t< T_actual >::ColsAtCompileTime)> * = nullptr>
T_actual &&	forward_as (T_actual &&a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , require_all_eigen_t< T_desired, T_actual > * = nullptr, std::enable_if_t< !std::is_same< value_type_t< T_actual >, value_type_t< T_desired >>::value||!internal::eigen_static_size_match(T_desired::RowsAtCompileTime, std::decay_t< T_actual >::RowsAtCompileTime)||!internal::eigen_static_size_match(T_desired::ColsAtCompileTime, std::decay_t< T_actual >::ColsAtCompileTime)> * = nullptr>
T_desired	forward_as (const T_actual &a)
	Assume which type we get. More...
template<typename T , typename... Ptrs, std::enable_if_t< sizeof...(Ptrs) > = 1>
Ptrs	holder (T &&arg, Ptrs *...pointers)
template<typename T >
T	holder (T &&arg)
template<typename F , typename... Args, require_not_plain_type_t< decltype(std::declval< F >()(std::declval< Args & >()...))> * = nullptr>
auto	make_holder (const F &func, Args &&...args)
	Constructs an expression from given arguments using given functor. More...
template<std::size_t N, class F >
constexpr auto	index_apply (F f)
	Calls given callable with an index sequence. More...
template<bool Condition, typename T1 , typename T2 , std::enable_if_t< Condition > * = nullptr>
T1	static_select (T1 &&a, T2 &&b)
	Returns one of the arguments that can be of different type, depending on the compile time condition. More...
template<bool Condition, typename T1 , typename T2 , std::enable_if_t<!Condition > * = nullptr>
T2	static_select (T1 &&a, T2 &&b)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_ccdf_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_prob > * = nullptr>
return_type_t< T_prob >	bernoulli_cdf (const T_n &n, const T_prob &theta)
	Returns the CDF of the Bernoulli distribution. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_cdf_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_prob > * = nullptr>
return_type_t< T_prob >	bernoulli_lccdf (const T_n &n, const T_prob &theta)
	Returns the log CCDF of the Bernoulli distribution. More...
template<typename T_n , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_prob > * = nullptr>
return_type_t< T_prob >	bernoulli_lcdf (const T_n &n, const T_prob &theta)
	Returns the log CDF of the Bernoulli distribution. More...
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_log (const T_n &n, const T_prob &theta)
template<typename T_y , typename T_prob >
return_type_t< T_prob >	bernoulli_log (const T_y &n, const T_prob &theta)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	bernoulli_logit_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	bernoulli_logit_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , require_eigen_t< T_x > * = nullptr>
return_type_t< T_x, T_alpha, T_beta >	bernoulli_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_beta, T_alpha >	bernoulli_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_x , typename T_alpha , typename T_beta , class RNG >
VectorBuilder< true, int, T_alpha >::type	bernoulli_logit_glm_rng (const T_x &x, const T_alpha &alpha, const T_beta &beta, RNG &rng)
	Returns a draw from the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_log (const T_n &n, const T_prob &theta)
template<bool propto, typename T_n , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_prob > * = nullptr>
return_type_t< T_prob >	bernoulli_logit_lpmf (const T_n &n, const T_prob &theta)
	Returns the log PMF of the logit-parametrized Bernoulli distribution. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_lpmf (const T_n &n, const T_prob &theta)
template<typename T_t , class RNG >
VectorBuilder< true, int, T_t >::type	bernoulli_logit_rng (const T_t &t, RNG &rng)
	Return a Bernoulli random variate with logit-parameterized chance of success using the specified random number generator. More...
template<bool propto, typename T_n , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_prob > * = nullptr>
return_type_t< T_prob >	bernoulli_lpmf (const T_n &n, const T_prob &theta)
	Returns the log PMF of the Bernoulli distribution. More...
template<typename T_y , typename T_prob >
return_type_t< T_prob >	bernoulli_lpmf (const T_y &n, const T_prob &theta)
template<typename T_theta , class RNG >
VectorBuilder< true, int, T_theta >::type	bernoulli_rng (const T_theta &theta, RNG &rng)
	Return a Bernoulli random variate with specified chance of success parameter using the specified random number generator. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_ccdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_cdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the CDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_cdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lccdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log CCDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lcdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log CDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_N, T_size1, T_size2 > * = nullptr>
return_type_t< T_size1, T_size2 >	beta_binomial_lpmf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log PMF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lpmf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_N , typename T_shape1 , typename T_shape2 , class RNG >
VectorBuilder< true, int, T_N, T_shape1, T_shape2 >::type	beta_binomial_rng (const T_N &N, const T_shape1 &alpha, const T_shape2 &beta, RNG &rng)
	Return a beta-binomial random variate with the specified population size, success, and failure parameters using the given random number generator. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_ccdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_cdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	Calculates the beta cumulative distribution function for the given variate and scale variables. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_cdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lccdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta_param)
	Returns the beta log complementary cumulative distribution function for the given probability, success, and failure parameters. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lcdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta_param)
	Returns the beta log cumulative distribution function for the given probability, success, and failure parameters. More...
template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale_succ, T_scale_fail > * = nullptr>
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lpdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	The log of the beta density for the specified scalar(s) given the specified sample stan::math::size(s). More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lpdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_ccdf_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_cdf_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lccdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log complementary cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lccdf(y | mu, kappa) = beta_lccdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lcdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lcdf(y | mu, kappa) = beta_lcdf(y | mu * kappa, (1 - mu) * kappa). More...
template<bool propto, typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<bool propto, typename T_y , typename T_loc , typename T_prec , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_prec > * = nullptr>
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lpdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	The log of the beta density for specified y, location, and precision: beta_proportion_lpdf(y | mu, kappa) = beta_lpdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lpdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<typename T_loc , typename T_prec , class RNG >
VectorBuilder< true, double, T_loc, T_prec >::type	beta_proportion_rng (const T_loc &mu, const T_prec &kappa, RNG &rng)
	Return a Beta random variate specified probability, location, and precision parameters: beta_proportion_rng(y | mu, kappa) = beta_rng(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_shape1 , typename T_shape2 , class RNG >
VectorBuilder< true, double, T_shape1, T_shape2 >::type	beta_rng (const T_shape1 &alpha, const T_shape2 &beta, RNG &rng)
	Return a Beta random variate with the supplied success and failure parameters using the given random number generator. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_ccdf_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_cdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the CDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_cdf_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lccdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log CCDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lcdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log CDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
template<bool propto, typename T_n , typename T_N , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_N, T_prob > * = nullptr>
return_type_t< T_prob >	binomial_logit_lpmf (const T_n &n, const T_N &N, const T_prob &alpha)
	Binomial log PMF in logit parametrization. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_lpmf (const T_n &n, const T_N &N, const T_prob &alpha)
template<bool propto, typename T_n , typename T_N , typename T_prob , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_N, T_prob > * = nullptr>
return_type_t< T_prob >	binomial_lpmf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log PMF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lpmf (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_N , typename T_theta , class RNG >
VectorBuilder< true, int, T_N, T_theta >::type	binomial_rng (const T_N &N, const T_theta &theta, RNG &rng)
	Return a pseudorandom binomial random variable for the given population size and chance of success parameters using the specified random number generator. More...
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	categorical_log (const T_n &ns, const T_prob &theta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	categorical_log (const T_n &ns, const T_prob &theta)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , require_eigen_t< T_x > * = nullptr, require_eigen_col_vector_t< T_alpha > * = nullptr, require_eigen_matrix_dynamic_t< T_beta > * = nullptr>
return_type_t< T_x, T_alpha, T_beta >	categorical_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with categorical distribution and logit (softmax) link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	categorical_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	categorical_logit_log (const T_n &ns, const T_prob &beta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	categorical_logit_log (const T_n &ns, const T_prob &beta)
template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_logit_lpmf (int n, const T_prob &beta)
template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_logit_lpmf (const std::vector< int > &ns, const T_prob &beta)
template<typename T_n , typename T_prob , require_st_integral< T_n > * = nullptr, require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_logit_lpmf (const T_n &ns, const T_prob &beta)
template<class RNG >
int	categorical_logit_rng (const Eigen::VectorXd &beta, RNG &rng)
	Return a draw from a Categorical distribution given a a vector of unnormalized log probabilities and a psuedo-random number generator. More...
template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_lpmf (int n, const T_prob &theta)
template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_lpmf (const std::vector< int > &ns, const T_prob &theta)
template<typename T_n , typename T_prob , require_st_integral< T_n > * = nullptr, require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	categorical_lpmf (const T_n &ns, const T_prob &theta)
template<class RNG >
int	categorical_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &theta, RNG &rng)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	cauchy_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	cauchy_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	cauchy_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	cauchy_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	cauchy_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Cauchy random variate for the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_ccdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_cdf (const T_y &y, const T_dof &nu)
	Returns the chi square cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_cdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lccdf (const T_y &y, const T_dof &nu)
	Returns the chi square log complementary cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lcdf (const T_y &y, const T_dof &nu)
	Returns the chi square log cumulative distribution function for the given variate and degrees of freedom. More...
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_log (const T_y &y, const T_dof &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_log (const T_y &y, const T_dof &nu)
template<bool propto, typename T_y , typename T_dof , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_dof > * = nullptr>
return_type_t< T_y, T_dof >	chi_square_lpdf (const T_y &y, const T_dof &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lpdf (const T_y &y, const T_dof &nu)
template<typename T_deg , class RNG >
VectorBuilder< true, double, T_deg >::type	chi_square_rng (const T_deg &nu, RNG &rng)
	Return a chi squared random variate with nu degrees of freedom using the specified random number generator. More...
template<bool propto, typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_log (const T_prob &theta, const T_prior_size &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_log (const T_prob &theta, const T_prior_size &alpha)
template<bool propto, typename T_prob , typename T_prior_size , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_prob, T_prior_size > * = nullptr>
return_type_t< T_prob, T_prior_size >	dirichlet_lpdf (const T_prob &theta, const T_prior_size &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpdf (const T_prob &theta, const T_prior_size &alpha)
template<bool propto, typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpmf (const T_prob &theta, const T_prior_size &alpha)
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpmf (const T_prob &theta, const T_prior_size &alpha)
template<class RNG >
Eigen::VectorXd	dirichlet_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &alpha, RNG &rng)
	Return a draw from a Dirichlet distribution with specified parameters and pseudo-random number generator. More...
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_ccdf_log (const T_y &y, const T_lower &lower, const T_upper &upper)
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_cdf (const T_y &y, const T_lower &lower, const T_upper &upper)
	Return the CDF of a discrete range distribution for the given y, lower and upper bounds (all integers). More...
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_cdf_log (const T_y &y, const T_lower &lower, const T_upper &upper)
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_lccdf (const T_y &y, const T_lower &lower, const T_upper &upper)
	Return the log CCDF of a discrete range distribution for the given y, lower and upper bounds (all integers). More...
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_lcdf (const T_y &y, const T_lower &lower, const T_upper &upper)
	Return the log CDF of a discrete range distribution for the given y, lower and upper bounds (all integers). More...
template<bool propto, typename T_y , typename T_lower , typename T_upper >
double	discrete_range_log (const T_y &y, const T_lower &lower, const T_upper &upper)
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_log (const T_y &y, const T_lower &lower, const T_upper &upper)
template<bool propto, typename T_y , typename T_lower , typename T_upper >
double	discrete_range_lpmf (const T_y &y, const T_lower &lower, const T_upper &upper)
	Return the log PMF of a discrete range for the given y, lower and upper bound (all integers). More...
template<typename T_y , typename T_lower , typename T_upper >
double	discrete_range_lpmf (const T_y &y, const T_lower &lower, const T_upper &upper)
template<typename T_lower , typename T_upper , class RNG >
VectorBuilder< true, int, T_lower, T_upper >::type	discrete_range_rng (const T_lower &lower, const T_upper &upper, RNG &rng)
	Return an integer random variate between the given lower and upper bounds (inclusive) using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	double_exponential_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	double_exponential_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log complementary cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	double_exponential_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log cumulative density function. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log probability density function. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	double_exponential_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a double exponential random variate with the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_loc , typename T_scale , typename T_inv_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_inv_scale >::type	exp_mod_normal_rng (const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda, RNG &rng)
	Return an exponentially modified normal random variate for the given location, scale, and inverse scale using the specified random number generator. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_ccdf_log (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>
return_type_t< T_y, T_inv_scale >	exponential_cdf (const T_y &y, const T_inv_scale &beta)
	Calculates the exponential cumulative distribution function for the given y and beta. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_cdf_log (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>
return_type_t< T_y, T_inv_scale >	exponential_lccdf (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>
return_type_t< T_y, T_inv_scale >	exponential_lcdf (const T_y &y, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_log (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_log (const T_y &y, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>
return_type_t< T_y, T_inv_scale >	exponential_lpdf (const T_y &y, const T_inv_scale &beta)
	The log of an exponential density for y with the specified inverse scale parameter. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_lpdf (const T_y &y, const T_inv_scale &beta)
template<typename T_inv , class RNG >
VectorBuilder< true, double, T_inv >::type	exponential_rng (const T_inv &beta, RNG &rng)
	Return a exponential random variate with inverse scale beta using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	frechet_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	frechet_lccdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	frechet_lcdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	frechet_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	frechet_rng (const T_shape &alpha, const T_scale &sigma, RNG &rng)
	Return a pseudorandom Frechet variate for the given shape and scale parameters using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_cdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The cumulative density function for a gamma distribution for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lccdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lcdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_inv_scale > * = nullptr>
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lpdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lpdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_shape , typename T_inv , class RNG >
VectorBuilder< true, double, T_shape, T_inv >::type	gamma_rng (const T_shape &alpha, const T_inv &beta, RNG &rng)
	Return a gamma random variate for the given shape and inverse scale parameters using the specified random number generator. More...
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, T_F, T_G, T_V, T_W, T_m0, T_C0 >	gaussian_dlm_obs_log (const T_y &y, const T_F &F, const T_G &G, const T_V &V, const T_W &W, const T_m0 &m0, const T_C0 &C0)
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, T_F, T_G, T_V, T_W, T_m0, T_C0 >	gaussian_dlm_obs_log (const T_y &y, const T_F &F, const T_G &G, const T_V &V, const T_W &W, const T_m0 &m0, const T_C0 &C0)
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 , require_all_eigen_matrix_dynamic_t< T_y, T_F, T_G, T_V, T_W, T_C0 > * = nullptr, require_eigen_col_vector_t< T_m0 > * = nullptr>
return_type_t< T_y, T_F, T_G, T_V, T_W, T_m0, T_C0 >	gaussian_dlm_obs_lpdf (const T_y &y, const T_F &F, const T_G &G, const T_V &V, const T_W &W, const T_m0 &m0, const T_C0 &C0)
	The log of a Gaussian dynamic linear model (GDLM). More...
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, T_F, T_G, T_V, T_W, T_m0, T_C0 >	gaussian_dlm_obs_lpdf (const T_y &y, const T_F &F, const T_G &G, const T_V &V, const T_W &W, const T_m0 &m0, const T_C0 &C0)
template<class RNG >
Eigen::MatrixXd	gaussian_dlm_obs_rng (const Eigen::MatrixXd &F, const Eigen::MatrixXd &G, const Eigen::MatrixXd &V, const Eigen::MatrixXd &W, const Eigen::VectorXd &m0, const Eigen::MatrixXd &C0, const int T, RNG &rng)
	Simulate random draw from Gaussian dynamic linear model (GDLM). More...
template<class RNG >
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	gaussian_dlm_obs_rng (const Eigen::MatrixXd &F, const Eigen::MatrixXd &G, const Eigen::VectorXd &V, const Eigen::MatrixXd &W, const Eigen::VectorXd &m0, const Eigen::MatrixXd &C0, const int T, RNG &rng)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	gumbel_cdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel distribution cumulative distribution for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_cdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	gumbel_lccdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log complementary cumulative distribution for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	gumbel_lcdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log cumulative distribution for the given location and scale. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	gumbel_lpdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log probability density for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_lpdf (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	gumbel_rng (const T_loc &mu, const T_scale &beta, RNG &rng)
	Return a Gumbel random variate with the given location and scale using the specified random number generator. More...
template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>
Eigen::MatrixXd	hmm_hidden_state_prob (const T_omega &log_omegas, const T_Gamma &Gamma, const T_rho &rho)
	For a hidden Markov model with observation y, hidden state x, and parameters theta, compute the marginal probability vector for each x, given y and theta, p(x_i | y, theta). More...
template<typename T_omega , typename T_Gamma , typename T_rho , class RNG , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>
std::vector< int >	hmm_latent_rng (const T_omega &log_omegas, const T_Gamma &Gamma, const T_rho &rho, RNG &rng)
	For a hidden Markov model with observation y, hidden state x, and parameters theta, generate samples from the posterior distribution of the hidden states, x. More...
template<typename T_omega , typename T_Gamma , typename T_rho , typename T_alpha >
auto	hmm_marginal_val (const Eigen::Matrix< T_omega, Eigen::Dynamic, Eigen::Dynamic > &omegas, const T_Gamma &Gamma_val, const T_rho &rho_val, Eigen::Matrix< T_alpha, Eigen::Dynamic, Eigen::Dynamic > &alphas, Eigen::Matrix< T_alpha, Eigen::Dynamic, 1 > &alpha_log_norms, T_alpha &norm_norm)
template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>
auto	hmm_marginal (const T_omega &log_omegas, const T_Gamma &Gamma, const T_rho &rho)
	For a Hidden Markov Model with observation y, hidden state x, and parameters theta, return the log marginal density, log p(y | theta). More...
template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_lpmf (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_lpmf (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<class RNG >
int	hypergeometric_rng (int N, int a, int b, RNG &rng)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_cdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_cdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lccdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square log complementary cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lcdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square log cumulative distribution function for the given variate and degrees of freedom. More...
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_log (const T_y &y, const T_dof &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_log (const T_y &y, const T_dof &nu)
template<bool propto, typename T_y , typename T_dof , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_dof > * = nullptr>
return_type_t< T_y, T_dof >	inv_chi_square_lpdf (const T_y &y, const T_dof &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lpdf (const T_y &y, const T_dof &nu)
template<typename T_deg , class RNG >
VectorBuilder< true, double, T_deg >::type	inv_chi_square_rng (const T_deg &nu, RNG &rng)
	Return a pseudorandom inverse chi squared variate with the nu degrees of freedom using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_cdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The CDF of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lccdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lcdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lpdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lpdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	inv_gamma_rng (const T_shape &alpha, const T_scale &beta, RNG &rng)
	Return a pseudorandom inverse gamma variate for the given shape and scale parameters using the specified random number generator. More...
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_log (const T_y &W, const T_dof &nu, const T_scale &S)
	The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_log (const T_y &W, const T_dof &nu, const T_scale &S)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_lpdf (const T_y &W, const T_dof &nu, const T_scale &S)
	The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_lpdf (const T_y &W, const T_dof &nu, const T_scale &S)
template<class RNG >
Eigen::MatrixXd	inv_wishart_rng (double nu, const Eigen::MatrixXd &S, RNG &rng)
template<bool propto, typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_log (const T_covar &L, const T_shape &eta)
template<typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_log (const T_covar &L, const T_shape &eta)
template<bool propto, typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_lpdf (const T_covar &L, const T_shape &eta)
template<typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_lpdf (const T_covar &L, const T_shape &eta)
template<class RNG >
Eigen::MatrixXd	lkj_corr_cholesky_rng (size_t K, double eta, RNG &rng)
template<bool propto, typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_log (const T_y &y, const T_shape &eta)
template<typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_log (const T_y &y, const T_shape &eta)
template<typename T_shape >
return_type_t< double, T_shape >	do_lkj_constant (const T_shape &eta, const unsigned int &K)
template<bool propto, typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_lpdf (const T_y &y, const T_shape &eta)
template<typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_lpdf (const T_y &y, const T_shape &eta)
template<class RNG >
Eigen::MatrixXd	lkj_corr_rng (size_t K, double eta, RNG &rng)
	Return a random correlation matrix (symmetric, positive definite, unit diagonal) of the specified dimensionality drawn from the LKJ distribution with the specified degrees of freedom using the specified random number generator. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_eigen_matrix_dynamic_t< T_y > * = nullptr, require_all_eigen_col_vector_t< T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	logistic_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	logistic_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	logistic_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	logistic_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	logistic_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Logistic random variate for the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	loglogistic_cdf (const T_y &y, const T_scale &alpha, const T_shape &beta)
	The loglogistic cumulative distribution function for the specified scalar(s) given the specified scales(s) and shape(s). More...
template<bool propto, typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	loglogistic_lpdf (const T_y &y, const T_scale &alpha, const T_shape &beta)
	The log of the loglogistic density for the specified scalar(s) given the specified scales(s) and shape(s). More...
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	loglogistic_lpdf (const T_y &y, const T_scale &alpha, const T_shape &beta)
template<typename T_scale , typename T_shape , class RNG >
VectorBuilder< true, double, T_scale, T_shape >::type	loglogistic_rng (const T_scale &alpha, const T_shape &beta, RNG &rng)
	Return a loglogistic random variate for the given scale and shape parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	lognormal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	lognormal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	lognormal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	lognormal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	lognormal_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a lognormal random variate for the given location and scale using the specified random number generator. More...
template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D , require_all_matrix_t< T_y, T_Mu, T_Sigma, T_D > * = nullptr>
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_log (const T_y &y, const T_Mu &Mu, const T_Sigma &Sigma, const T_D &D)
	The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D , require_all_matrix_t< T_y, T_Mu, T_Sigma, T_D > * = nullptr>
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_log (const T_y &y, const T_Mu &Mu, const T_Sigma &Sigma, const T_D &D)
template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D , require_all_matrix_t< T_y, T_Mu, T_Sigma, T_D > * = nullptr>
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_lpdf (const T_y &y, const T_Mu &Mu, const T_Sigma &Sigma, const T_D &D)
	The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D , require_all_matrix_t< T_y, T_Mu, T_Sigma, T_D > * = nullptr>
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_lpdf (const T_y &y, const T_Mu &Mu, const T_Sigma &Sigma, const T_D &D)
template<class RNG >
Eigen::MatrixXd	matrix_normal_prec_rng (const Eigen::MatrixXd &Mu, const Eigen::MatrixXd &Sigma, const Eigen::MatrixXd &D, RNG &rng)
	Sample from the the matrix normal distribution for the given Mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_log (const T_y &y, const T_covar &L, const T_w &w)
	The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_log (const T_y &y, const T_covar &L, const T_w &w)
template<bool propto, typename T_y , typename T_covar , typename T_w , require_all_eigen_matrix_dynamic_t< T_y, T_covar > * = nullptr, require_eigen_col_vector_t< T_w > * = nullptr>
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_lpdf (const T_y &y, const T_covar &L, const T_w &w)
	The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_lpdf (const T_y &y, const T_covar &L, const T_w &w)
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_log (const T_y &y, const T_covar &Sigma, const T_w &w)
	The log of a multivariate Gaussian Process for the given y, Sigma, and w. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_log (const T_y &y, const T_covar &Sigma, const T_w &w)
template<bool propto, typename T_y , typename T_covar , typename T_w , require_all_matrix_t< T_y, T_covar > * = nullptr, require_col_vector_t< T_w > * = nullptr>
return_type_t< T_y, T_covar, T_w >	multi_gp_lpdf (const T_y &y, const T_covar &Sigma, const T_w &w)
	The log of a multivariate Gaussian Process for the given y, Sigma, and w. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_lpdf (const T_y &y, const T_covar &Sigma, const T_w &w)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
template<bool propto, typename T_y , typename T_loc , typename T_covar , require_any_not_vector_vt< is_stan_scalar, T_y, T_loc > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_covar > * = nullptr>
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_lpdf (const T_y &y, const T_loc &mu, const T_covar &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_lpdf (const T_y &y, const T_loc &mu, const T_covar &L)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_cholesky_rng (const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &L, RNG &rng)
	Return a multivariate normal random variate with the given location and Cholesky factorization of the covariance using the specified random number generator. More...
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_prec_rng (const T_loc &mu, const Eigen::MatrixXd &S, RNG &rng)
	Return a multivariate normal random variate with the given location and precision using the specified random number generator. More...
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_rng (const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
	Return a multivariate normal random variate with the given location and covariance using the specified random number generator. More...
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
	Return the log of the multivariate Student t distribution at the specified arguments. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
	Return the log of the multivariate Student t distribution at the specified arguments. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_student_t_rng (double nu, const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
	Return a multivariate student-t random variate with the given degrees of freedom location and covariance using the specified random number generator. More...
template<bool propto, typename T_prob >
return_type_t< T_prob >	multinomial_log (const std::vector< int > &ns, const T_prob &theta)
template<typename T_prob >
return_type_t< T_prob >	multinomial_log (const std::vector< int > &ns, const T_prob &theta)
template<bool propto, typename T_beta , require_eigen_col_vector_t< T_beta > * = nullptr>
return_type_t< T_beta >	multinomial_logit_log (const std::vector< int > &ns, const T_beta &beta)
template<typename T_beta , require_eigen_col_vector_t< T_beta > * = nullptr>
return_type_t< T_beta >	multinomial_logit_log (const std::vector< int > &ns, const T_beta &beta)
template<bool propto, typename T_beta , typename T_prob = scalar_type_t<T_beta>, require_eigen_col_vector_t< T_beta > * = nullptr>
return_type_t< T_prob >	multinomial_logit_lpmf (const std::vector< int > &ns, const T_beta &beta)
	Multinomial log PMF in log parametrization. More...
template<typename T_beta , require_eigen_col_vector_t< T_beta > * = nullptr>
return_type_t< T_beta >	multinomial_logit_lpmf (const std::vector< int > &ns, const T_beta &beta)
template<class RNG , typename T_beta , require_eigen_col_vector_t< T_beta > * = nullptr>
std::vector< int >	multinomial_logit_rng (const T_beta &beta, int N, RNG &rng)
	Return a draw from a Multinomial distribution given a a vector of unnormalized log probabilities and a pseudo-random number generator. More...
template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>
return_type_t< T_prob >	multinomial_lpmf (const std::vector< int > &ns, const T_prob &theta)
template<typename T_prob >
return_type_t< T_prob >	multinomial_lpmf (const std::vector< int > &ns, const T_prob &theta)
template<class T_theta , class RNG , require_eigen_col_vector_t< T_theta > * = nullptr>
std::vector< int >	multinomial_rng (const T_theta &theta, int N, RNG &rng)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_ccdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_cdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_cdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lccdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lcdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<bool propto, typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision , require_eigen_t< T_x > * = nullptr>
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
	Returns the log PMF of the Generalized Linear Model (GLM) with Negative-Binomial-2 distribution and log link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<bool propto, typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<bool propto, typename T_n , typename T_log_location , typename T_precision , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_log_location, T_precision > * = nullptr>
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_lpmf (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_lpmf (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_loc , typename T_inv , class RNG >
VectorBuilder< true, int, T_loc, T_inv >::type	neg_binomial_2_log_rng (const T_loc &eta, const T_inv &phi, RNG &rng)
	Return a negative binomial random variate with the specified log-location and inverse dispersion parameters using the given random number generator. More...
template<bool propto, typename T_n , typename T_location , typename T_precision , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_location, T_precision > * = nullptr>
return_type_t< T_location, T_precision >	neg_binomial_2_lpmf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lpmf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_loc , typename T_prec , class RNG >
VectorBuilder< true, int, T_loc, T_prec >::type	neg_binomial_2_rng (const T_loc &mu, const T_prec &phi, RNG &rng)
	Return a negative binomial random variate with the specified location and precision parameters using the given random number generator. More...
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_ccdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_cdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_cdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lccdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta_param)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lcdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta_param)
template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_n , typename T_shape , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_shape, T_inv_scale > * = nullptr>
return_type_t< T_shape, T_inv_scale >	neg_binomial_lpmf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lpmf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_shape , typename T_inv , class RNG >
VectorBuilder< true, int, T_shape, T_inv >::type	neg_binomial_rng (const T_shape &alpha, const T_inv &beta, RNG &rng)
	Return a negative binomial random variate with the specified shape and inverse scale parameters using the given random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Calculates the normal cumulative distribution function for the given variate, location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale , require_eigen_t< T_x > * = nullptr>
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
	Returns the log PDF of the Generalized Linear Model (GLM) with Normal distribution and id link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_loc, T_scale >	normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	normal_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Normal random variate for the given location and scale using the specified random number generator. More...
template<bool propto, typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_log (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_log (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_lpdf (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_lpdf (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_x , typename T_beta , typename T_cuts , require_eigen_t< T_x > * = nullptr, require_all_eigen_col_vector_t< T_beta, T_cuts > * = nullptr>
return_type_t< T_x, T_beta, T_cuts >	ordered_logistic_glm_lpmf (const T_y &y, const T_x &x, const T_beta &beta, const T_cuts &cuts)
	Returns the log PMF of the ordinal regression Generalized Linear Model (GLM). More...
template<typename T_y , typename T_x , typename T_beta , typename T_cuts >
return_type_t< T_x, T_beta, T_cuts >	ordered_logistic_glm_lpmf (const T_y &y, const T_x &x, const T_beta &beta, const T_cuts &cuts)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_log (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the integer/s given the vector of continuous location/s and specified cutpoints in an ordered logistic model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_log (const T_y &y, const T_loc &lambda, const T_cut &c)
template<bool propto, typename T_y , typename T_loc , typename T_cut , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_cut > * = nullptr>
return_type_t< T_loc, T_cut >	ordered_logistic_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and specified cutpoints in an ordered logistic model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
template<class RNG >
int	ordered_logistic_rng (double eta, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &c, RNG &rng)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_log (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the integer/s given the vector of continuous location/s and specified cutpoints in an ordered probit model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_log (const T_y &y, const T_loc &lambda, const T_cut &c)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and array of specified cutpoints in an ordered probit model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
template<class RNG >
int	ordered_probit_rng (double eta, const Eigen::VectorXd &c, RNG &rng)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_ccdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	pareto_cdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_cdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	pareto_lccdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	pareto_lcdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<bool propto, typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<bool propto, typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_scale, T_shape >	pareto_lpdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_lpdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	pareto_rng (const T_scale &y_min, const T_shape &alpha, RNG &rng)
	Return a Pareto random variate for the given shape and scale parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_cdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_cdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lccdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lcdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lpdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lpdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_loc , typename T_scale , typename T_shape , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_shape >::type	pareto_type_2_rng (const T_loc &mu, const T_scale &lambda, const T_shape &alpha, RNG &rng)
	Return a Pareto type 2 random variate for the given location, scale, and shape using the specified random number generator. More...
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_ccdf_log (const T_y &y, const T_theta &theta)
template<bool propto, typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_cdf (const T_y &y, const T_theta &theta)
	Returns the CDF for the Poisson-binomial distribution evaluated at the specified number of successes and probabilities of successes. More...
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_cdf (const T_y &y, const T_theta &theta)
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_cdf_log (const T_y &y, const T_theta &theta)
template<bool propto, typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lccdf (const T_y &y, const T_theta &theta)
	Returns the log CCDF for the Poisson-binomial distribution evaluated at the specified number of successes and probabilities of successes. More...
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lccdf (const T_y &y, const T_theta &theta)
template<bool propto, typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lcdf (const T_y &y, const T_theta &theta)
	Returns the log CDF for the Poisson-binomial distribution evaluated at the specified number of successes and probabilities of successes. More...
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lcdf (const T_y &y, const T_theta &theta)
template<bool propto, typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_log (const T_y &y, const T_theta &theta)
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_log (const T_y &y, const T_theta &theta)
template<bool propto, typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lpmf (const T_y &y, const T_theta &theta)
	Returns the log PMF for the Poisson-binomial distribution evaluated at an specified array of numbers of successes and probabilities of successes. More...
template<typename T_y , typename T_theta >
return_type_t< T_theta >	poisson_binomial_lpmf (const T_y &y, const T_theta &theta)
template<typename T_theta , typename RNG , require_eigen_vt< std::is_arithmetic, T_theta > * = nullptr>
int	poisson_binomial_rng (const T_theta &theta, RNG &rng)
	Return a pseudorandom Poisson binomial random variable for the given vector of success parameters using the specified random number generator. More...
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_ccdf_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_cdf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_cdf_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lccdf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lcdf (const T_n &n, const T_rate &lambda)
template<bool propto, typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_log (const T_n &n, const T_rate &lambda)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , require_eigen_t< T_x > * = nullptr>
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Poisson distribution and log link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_log (const T_n &n, const T_log_rate &alpha)
template<typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_log (const T_n &n, const T_log_rate &alpha)
template<bool propto, typename T_n , typename T_log_rate , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_log_rate > * = nullptr>
return_type_t< T_log_rate >	poisson_log_lpmf (const T_n &n, const T_log_rate &alpha)
template<typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_lpmf (const T_n &n, const T_log_rate &alpha)
template<typename T_rate , class RNG >
VectorBuilder< true, int, T_rate >::type	poisson_log_rng (const T_rate &alpha, RNG &rng)
	Return a Poisson random variate with specified log rate parameter using the given random number generator. More...
template<bool propto, typename T_n , typename T_rate , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_rate > * = nullptr>
return_type_t< T_rate >	poisson_lpmf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lpmf (const T_n &n, const T_rate &lambda)
template<typename T_rate , class RNG >
VectorBuilder< true, int, T_rate >::type	poisson_rng (const T_rate &lambda, RNG &rng)
	Return a Poisson random variate with specified rate parameter using the given random number generator. More...
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_ccdf_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>
return_type_t< T_y, T_scale >	rayleigh_cdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_cdf_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>
return_type_t< T_y, T_scale >	rayleigh_lccdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>
return_type_t< T_y, T_scale >	rayleigh_lcdf (const T_y &y, const T_scale &sigma)
template<bool propto, typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_log (const T_y &y, const T_scale &sigma)
template<bool propto, typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>
return_type_t< T_y, T_scale >	rayleigh_lpdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_lpdf (const T_y &y, const T_scale &sigma)
template<typename T_scale , class RNG >
VectorBuilder< true, double, T_scale >::type	rayleigh_rng (const T_scale &sigma, RNG &rng)
	Return a Rayleigh random variate with scale parameter sigma using the specified random number generator. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_cdf (const T_y &y, const T_dof &nu, const T_scale &s)
	The CDF of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_cdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lccdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lcdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<bool propto, typename T_y , typename T_dof , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_dof, T_scale > * = nullptr>
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lpdf (const T_y &y, const T_dof &nu, const T_scale &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lpdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_deg , typename T_scale , class RNG >
VectorBuilder< true, double, T_deg, T_scale >::type	scaled_inv_chi_square_rng (const T_deg &nu, const T_scale &s, RNG &rng)
	Return a scaled chi square random variate for the given number of degrees of freedom and scale using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness >
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_skewness > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
	Returns the skew double exponential cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness >
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_skewness > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
	Returns the skew double exponential log complementary cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_skewness > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
	Returns the skew double exponential log cumulative density function. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_skewness >
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness >
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_skewness , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_skewness > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
	Returns the skew double exponential log probability density function. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_skewness >
return_type_t< T_y, T_loc, T_scale, T_skewness >	skew_double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_skewness &tau)
template<typename T_loc , typename T_scale , typename T_skewness , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_skewness >::type	skew_double_exponential_rng (const T_loc &mu, const T_scale &sigma, const T_skewness &tau, RNG &rng)
	Return a skew double exponential random variate with the given location scale and skewness using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_loc , typename T_scale , typename T_shape , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_shape >::type	skew_normal_rng (const T_loc &mu, const T_scale &sigma, const T_shape &alpha, RNG &rng)
	Return a Skew-normal random variate for the given location, scale, and shape using the specified random number generator. More...
template<typename T_y >
return_type_t< T_y >	std_normal_ccdf_log (const T_y &y)
template<typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>
return_type_t< T_y >	std_normal_cdf (const T_y &y)
	Calculates the standard normal cumulative distribution function for the given variate. More...
template<typename T_y >
return_type_t< T_y >	std_normal_cdf_log (const T_y &y)
template<typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>
return_type_t< T_y >	std_normal_lccdf (const T_y &y)
template<typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>
return_type_t< T_y >	std_normal_lcdf (const T_y &y)
template<bool propto, typename T_y >
return_type_t< T_y >	std_normal_log (const T_y &y)
	The log of a standard normal density for the specified scalar(s). More...
template<typename T_y >
return_type_t< T_y >	std_normal_log (const T_y &y)
template<bool propto, typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>
return_type_t< T_y >	std_normal_lpdf (const T_y &y)
	The log of the normal density for the specified scalar(s) given a location of 0 and a scale of 1. More...
template<typename T_y >
return_type_t< T_y >	std_normal_lpdf (const T_y &y)
template<class RNG >
double	std_normal_rng (RNG &rng)
	Return a standard Normal random variate using the specified random number generator. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_ccdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_cdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_cdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lccdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lcdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_dof, T_loc, T_scale > * = nullptr>
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_deg , typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_deg, T_loc, T_scale >::type	student_t_rng (const T_deg &nu, const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a student-t random variate for the given degrees of freedom, location, and scale using the specified random number generator. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_ccdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>
return_type_t< T_y, T_low, T_high >	uniform_cdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_cdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>
return_type_t< T_y, T_low, T_high >	uniform_lccdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>
return_type_t< T_y, T_low, T_high >	uniform_lcdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<bool propto, typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
	The log of a uniform density for the given y, lower, and upper bound. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<bool propto, typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>
return_type_t< T_y, T_low, T_high >	uniform_lpdf (const T_y &y, const T_low &alpha, const T_high &beta)
	The log of a uniform density for the given y, lower, and upper bound. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_lpdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_alpha , typename T_beta , class RNG >
VectorBuilder< true, double, T_alpha, T_beta >::type	uniform_rng (const T_alpha &alpha, const T_beta &beta, RNG &rng)
	Return a uniform random variate for the given upper and lower bounds using the specified random number generator. More...
template<typename T_x , typename T_mu , typename T_k >
return_type_t< T_x, T_mu, T_k >	von_mises_cdf (const T_x &x, const T_mu &mu, const T_k &k)
	Calculates the cumulative distribution function of the von Mises distribution: More...
template<typename T_x , typename T_mu , typename T_k >
return_type_t< T_x, T_mu, T_k >	von_mises_lccdf (const T_x &x, const T_mu &mu, const T_k &k)
	Calculates the log of the complement of the cumulative distribution function of the von Mises distribution: More...
template<typename T_x , typename T_mu , typename T_k >
return_type_t< T_x, T_mu, T_k >	von_mises_lcdf (const T_x &x, const T_mu &mu, const T_k &k)
	Calculates the log of the cumulative distribution function of the von Mises distribution: More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_lpdf (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_lpdf (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_loc , typename T_conc , class RNG >
VectorBuilder< true, double, T_loc, T_conc >::type	von_mises_rng (const T_loc &mu, const T_conc &kappa, RNG &rng)
	Return a von Mises random variate for the given location and concentration using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	weibull_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull cumulative distribution function for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	weibull_lccdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log complementary cumulative distribution function for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	weibull_lcdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log cumulative distribution function for the given location and scale. More...
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>
return_type_t< T_y, T_shape, T_scale >	weibull_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log probability density for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	weibull_rng (const T_shape &alpha, const T_scale &sigma, RNG &rng)
	Return a Weibull random variate for the given shape and scale parameters using the specified random number generator. More...
template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
	The log of the first passage time density function for a (Wiener) drift diffusion model for the given \(y\), boundary separation \(\alpha\), nondecision time \(\tau\), relative bias \(\beta\), and drift rate \(\delta\). More...
template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_lpdf (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
	The log of the first passage time density function for a (Wiener) drift diffusion model for the given \(y\), boundary separation \(\alpha\), nondecision time \(\tau\), relative bias \(\beta\), and drift rate \(\delta\). More...
template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_lpdf (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_log (const T_y &W, const T_dof &nu, const T_scale &S)
	The log of the Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_log (const T_y &W, const T_dof &nu, const T_scale &S)
template<bool propto, typename T_y , typename T_dof , typename T_scale , require_stan_scalar_t< T_dof > * = nullptr, require_all_matrix_t< T_y, T_scale > * = nullptr>
return_type_t< T_y, T_dof, T_scale >	wishart_lpdf (const T_y &W, const T_dof &nu, const T_scale &S)
	The log of the Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_lpdf (const T_y &W, const T_dof &nu, const T_scale &S)
template<class RNG >
Eigen::MatrixXd	wishart_rng (double nu, const Eigen::MatrixXd &S, RNG &rng)
template<typename... Pargs>
double *	accumulate_adjoints (double *dest, const var &x, Pargs &&...args)
	Accumulate adjoints from x into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer. More...
template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr, typename... Pargs>
double *	accumulate_adjoints (double *dest, VarVec &&x, Pargs &&...args)
	Accumulate adjoints from std::vector x into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer. More...
template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr, typename... Pargs>
double *	accumulate_adjoints (double *dest, VecContainer &&x, Pargs &&...args)
	Accumulate adjoints from x (a std::vector of containers containing vars) into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer. More...
template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr, typename... Pargs>
double *	accumulate_adjoints (double *dest, EigT &&x, Pargs &&...args)
	Accumulate adjoints from x (an Eigen type containing vars) into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer. More...
template<typename Arith , require_st_arithmetic< Arith > * = nullptr, typename... Pargs>
double *	accumulate_adjoints (double *dest, Arith &&x, Pargs &&...args)
	Ignore arithmetic types. More...
double *	accumulate_adjoints (double *dest)
	End accumulate_adjoints recursion and return pointer. More...
template<int R, int C>
vari **	build_vari_array (const Eigen::Matrix< var, R, C > &x)
	Allocates and populates a flat array of vari pointers in the autodiff arena with the varis pointed to by the vars in the input Eigen matrix. More...
template<typename T , typename F >
internal::callback_vari< plain_type_t< T >, F > *	make_callback_vari (T &&value, F &&functor)
	Creates a new vari with given value and a callback that implements the reverse pass (chain). More...
template<typename T , typename F >
var_value< plain_type_t< T > >	make_callback_var (T &&value, F &&functor)
	Creates a new var initialized with a callback_vari with a given value and reverse-pass callback functor. More...
template<typename T >
auto	make_chainable_ptr (T &&obj)
	Store the given object in a chainable_object so it is destructed only when the chainable stack memory is recovered and return a pointer to the underlying object. More...
template<typename T >
auto	make_unsafe_chainable_ptr (T &&obj)
	Store the given object in a chainable_object so it is destructed only when the chainable stack memory is recovered and return a pointer to the underlying object This function differs from make_chainable_object in that this class does not evaluate expressions. More...
template<typename... Pargs>
size_t	count_vars (Pargs &&...args)
	Count the number of vars in the input argument list. More...
template<typename Arith , typename = require_arithmetic_t<scalar_type_t<Arith>>>
Arith	deep_copy_vars (Arith &&arg)
	Forward arguments that do not contain vars. More...
auto	deep_copy_vars (const var &arg)
	Copy the value of a var but reallocate a new vari. More...
template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr>
auto	deep_copy_vars (VarVec &&arg)
	Copy the vars in arg but reallocate new varis for them. More...
template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr>
auto	deep_copy_vars (VecContainer &&arg)
	Copy the vars in arg but reallocate new varis for them. More...
template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr>
auto	deep_copy_vars (EigT &&arg)
	Copy the vars in arg but reallocate new varis for them. More...
static bool	empty_nested ()
	Return true if there is no nested autodiff being executed. More...
static void	grad ()
	Compute the gradient for all variables starting from the end of the AD tape. More...
template<typename Vari >
static void	grad (Vari *vi)
	Compute the gradient for all variables starting from the specified root variable implementation. More...
static size_t	nested_size ()
var	operator+ (const var &a, const var &b)
	Addition operator for variables (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator+ (const var &a, Arith b)
	Addition operator for variable and scalar (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator+ (Arith a, const var &b)
	Addition operator for scalar and variable (C++). More...
template<typename VarMat1 , typename VarMat2 , require_all_rev_matrix_t< VarMat1, VarMat2 > * = nullptr>
auto	add (const VarMat1 &a, const VarMat2 &b)
	Addition operator for matrix variables (C++). More...
template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	add (const VarMat &a, const Arith &b)
	Addition operator for a matrix variable and arithmetic (C++). More...
template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	add (const Arith &a, const VarMat &b)
	Addition operator for an arithmetic type and matrix variable (C++). More...
template<typename Var , typename EigMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr>
auto	add (const Var &a, const EigMat &b)
	Addition operator for an arithmetic matrix and variable (C++). More...
template<typename EigMat , typename Var , require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>
auto	add (const EigMat &a, const Var &b)
	Addition operator for a variable and arithmetic matrix (C++). More...
template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	add (const Var &a, const VarMat &b)
	Addition operator for a variable and variable matrix (C++). More...
template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	add (const VarMat &a, const Var &b)
	Addition operator for a variable matrix and variable (C++). More...
template<typename T1 , typename T2 , require_any_var_vt< std::is_arithmetic, T1, T2 > * = nullptr, require_any_arithmetic_t< T1, T2 > * = nullptr>
auto	add (const T1 &a, const T2 &b)
template<typename T1 , typename T2 , require_all_var_vt< std::is_arithmetic, T1, T2 > * = nullptr>
auto	add (const T1 &a, const T2 &b)
template<typename VarMat1 , typename VarMat2 , require_any_var_matrix_t< VarMat1, VarMat2 > * = nullptr>
auto	operator+ (const VarMat1 &a, const VarMat2 &b)
	Addition operator for matrix variables. More...
var	operator/ (const var &dividend, const var &divisor)
	Division operator for two variables (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator/ (const var &dividend, Arith divisor)
	Division operator for dividing a variable by a scalar (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator/ (Arith dividend, const var &divisor)
	Division operator for dividing a scalar by a variable (C++). More...
template<typename Scalar , typename Mat , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr, require_all_st_var_or_arithmetic< Scalar, Mat > * = nullptr, require_any_st_var< Scalar, Mat > * = nullptr>
auto	divide (const Mat &m, Scalar c)
	Return matrix divided by scalar. More...
template<typename Scalar , typename Mat , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr, require_all_st_var_or_arithmetic< Scalar, Mat > * = nullptr, require_any_st_var< Scalar, Mat > * = nullptr>
auto	divide (Scalar c, const Mat &m)
	Return scalar divided by matrix. More...
template<typename Mat1 , typename Mat2 , require_all_matrix_st< is_var_or_arithmetic, Mat1, Mat2 > * = nullptr, require_any_matrix_st< is_var, Mat1, Mat2 > * = nullptr>
auto	divide (const Mat1 &m1, const Mat2 &m2)
	Return a matrix divided by a matrix elementwise. More...
template<typename T1 , typename T2 , require_any_var_matrix_t< T1, T2 > * = nullptr>
auto	operator/ (const T1 &dividend, const T2 &divisor)
std::complex< var >	operator/ (const std::complex< var > &x1, const std::complex< var > &x2)
bool	operator== (const var &a, const var &b)
	Equality operator comparing two variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator== (const var &a, Arith b)
	Equality operator comparing a variable's value and a double (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator== (Arith a, const var &b)
	Equality operator comparing a scalar and a variable's value (C++). More...
bool	operator== (const var &x, const std::complex< var > &z)
	Return true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero. More...
bool	operator== (const std::complex< var > &z, const var &y)
	Return true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero. More...
bool	operator> (const var &a, const var &b)
	Greater than operator comparing variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator> (const var &a, Arith b)
	Greater than operator comparing variable's value and double (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator> (Arith a, const var &b)
	Greater than operator comparing a double and a variable's value (C++). More...
bool	operator>= (const var &a, const var &b)
	Greater than or equal operator comparing two variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator>= (const var &a, Arith b)
	Greater than or equal operator comparing variable's value and double (C++). More...
template<typename Arith , typename Var , require_arithmetic_t< Arith > * = nullptr>
bool	operator>= (Arith a, const var &b)
	Greater than or equal operator comparing double and variable's value (C++). More...
bool	operator< (const var &a, const var &b)
	Less than operator comparing variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator< (const var &a, Arith b)
	Less than operator comparing variable's value and a double (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator< (Arith a, const var &b)
	Less than operator comparing a double and variable's value (C++). More...
bool	operator<= (const var &a, const var &b)
	Less than or equal operator comparing two variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator<= (const var &a, Arith b)
	Less than or equal operator comparing a variable's value and a scalar (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator<= (Arith a, const var &b)
	Less than or equal operator comparing a double and variable's value (C++). More...
bool	operator&& (const var &x, const var &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator&& (const var &x, Arith y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator&& (Arith x, const var &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
bool	operator|| (const var &x, const var &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator|| (const var &x, Arith y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator|| (Arith x, const var &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
var	operator* (const var &a, const var &b)
	Multiplication operator for two variables (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator* (const var &a, Arith b)
	Multiplication operator for a variable and a scalar (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator* (Arith a, const var &b)
	Multiplication operator for a scalar and a variable (C++). More...
bool	operator!= (const var &a, const var &b)
	Inequality operator comparing two variables' values (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator!= (const var &a, Arith b)
	Inequality operator comparing a variable's value and a double (C++). More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
bool	operator!= (Arith a, const var &b)
	Inequality operator comparing a double and a variable's value (C++). More...
bool	operator!= (const var &x, const std::complex< var > &z)
	Return false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero. More...
bool	operator!= (const std::complex< var > &z, const var &y)
	Return false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero. More...
var	operator- (const var &a, const var &b)
	Subtraction operator for variables. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator- (const var &a, Arith b)
	Subtraction operator for variable and scalar. More...
template<typename Arith , require_arithmetic_t< Arith > * = nullptr>
var	operator- (Arith a, const var &b)
	Subtraction operator for scalar and variable. More...
template<typename VarMat1 , typename VarMat2 , require_all_rev_matrix_t< VarMat1, VarMat2 > * = nullptr>
auto	subtract (const VarMat1 &a, const VarMat2 &b)
	Subtraction operator for matrix variables. More...
template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	subtract (const VarMat &a, const Arith &b)
	Subtraction operator for a matrix variable and arithmetic. More...
template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	subtract (const Arith &a, const VarMat &b)
	Subtraction operator for an arithmetic type and matrix variable. More...
template<typename Var , typename EigMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr>
auto	subtract (const Var &a, const EigMat &b)
	Subtraction operator for an arithmetic matrix and variable. More...
template<typename EigMat , typename Var , require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>
auto	subtract (const EigMat &a, const Var &b)
	Subtraction operator for a variable and arithmetic matrix. More...
template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>
auto	subtract (const Var &a, const VarMat &b)
	Subtraction operator for a variable and variable matrix. More...
template<typename Var , typename VarMat , require_rev_matrix_t< VarMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>
auto	subtract (const VarMat &a, const Var &b)
	Subtraction operator for a variable matrix and variable. More...
template<typename T1 , typename T2 , require_any_var_vt< std::is_arithmetic, T1, T2 > * = nullptr, require_any_arithmetic_t< T1, T2 > * = nullptr>
auto	subtract (const T1 &a, const T2 &b)
template<typename T1 , typename T2 , require_all_var_vt< std::is_arithmetic, T1, T2 > * = nullptr>
auto	subtract (const T1 &a, const T2 &b)
template<typename VarMat1 , typename VarMat2 , require_any_var_matrix_t< VarMat1, VarMat2 > * = nullptr>
auto	operator- (const VarMat1 &a, const VarMat2 &b)
	Addition operator for matrix variables. More...
var &	operator-- (var &a)
	Prefix decrement operator for variables (C++). More...
var	operator-- (var &a, int)
	Postfix decrement operator for variables (C++). More...
var &	operator++ (var &a)
	Prefix increment operator for variables (C++). More...
var	operator++ (var &a, int)
	Postfix increment operator for variables (C++). More...
var	operator- (const var &a)
	Unary negation operator for variables (C++). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	operator- (const T &a)
	Compute additive inverse of input. More...
bool	operator! (const var &x)
	Return the negation of the value of the argument as defined by !. More...
var	operator+ (const var &a)
	Unary plus operator for variables (C++). More...
template<typename Arith , typename VecVar , typename VecArith , typename... ContainerOperands, typename... ContainerGradients>
var	precomputed_gradients (Arith value, const VecVar &operands, const VecArith &gradients, const std::tuple< ContainerOperands... > &container_operands=std::tuple<>(), const std::tuple< ContainerGradients... > &container_gradients=std::tuple<>())
	This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands. More...
void	print_stack (std::ostream &o)
	Prints the autodiff variable stack. More...
template<typename EigVar , typename EigVari , typename EigDbl >
void	read_vi_val_adj (const EigVar &VarMat, EigVari &VariMat, EigDbl &ValMat, EigDbl &AdjMat)
	Function applying the vi_val_adj_functor to extract the vari*, values, and adjoints of a given var matrix into separate matrices. More...
template<typename EigRev , typename EigDbl >
void	read_val_adj (const EigRev &VarMat, EigDbl &ValMat, EigDbl &AdjMat)
	Function applying the val_adj_functor to extract the values and adjoints of a given var or vari matrix into separate matrices. More...
template<typename EigVar , typename EigVari , typename EigDbl >
void	read_vi_val (const EigVar &VarMat, EigVari &VariMat, EigDbl &ValMat)
	Function applying the vi_val_functor to extract the varis and and values of a given var matrix into separate matrices. More...
template<typename EigVar , typename EigVari , typename EigDbl >
void	read_vi_adj (const EigVar &VarMat, EigVari &VariMat, EigDbl &AdjMat)
	Function applying the vi_adj_functor to extract the varis and and adjoints of a given var matrix into separate matrices. More...
static void	recover_memory ()
	Recover memory used for all variables for reuse. More...
static void	recover_memory_nested ()
	Recover only the memory used for the top nested call. More...
template<typename F >
void	reverse_pass_callback (F &&functor)
	Puts a callback on the autodiff stack to be called in reverse pass. More...
template<typename... Pargs>
vari **	save_varis (vari **dest, const var &x, Pargs &&...args)
	Save the vari pointer in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer. More...
template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr, typename... Pargs>
vari **	save_varis (vari **dest, VarVec &&x, Pargs &&...args)
	Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer. More...
template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr, typename... Pargs>
vari **	save_varis (vari **dest, VecContainer &&x, Pargs &&...args)
	Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer. More...
template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr, typename... Pargs>
vari **	save_varis (vari **dest, EigT &&x, Pargs &&...args)
	Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer. More...
template<typename Arith , require_st_arithmetic< Arith > * = nullptr, typename... Pargs>
vari **	save_varis (vari **dest, Arith &&x, Pargs &&...args)
	Ignore arithmetic types. More...
vari **	save_varis (vari **dest)
	End save_varis recursion and return pointer. More...
static void	set_zero_all_adjoints ()
	Reset all adjoint values in the stack to zero. More...
static void	set_zero_all_adjoints_nested ()
	Reset all adjoint values in the top nested portion of the stack to zero. More...
static void	start_nested ()
	Record the current position so that recover_memory_nested() can find it. More...
template<typename Vari >
static void	grad (Vari *vi)
void	zero_adjoints () noexcept
	End of recursion for set_zero_adjoints. More...
template<typename T , require_st_arithmetic< T > * = nullptr>
void	zero_adjoints (T &x) noexcept
	Do nothing for non-autodiff arguments. More...
void	zero_adjoints (var &x)
	Zero the adjoint of the vari in the first argument. More...
template<typename EigMat , require_eigen_vt< is_autodiff, EigMat > * = nullptr>
void	zero_adjoints (EigMat &x)
	Zero the adjoints of the varis of every var in an Eigen::Matrix container. More...
template<typename StdVec , require_std_vector_st< is_autodiff, StdVec > * = nullptr>
void	zero_adjoints (StdVec &x)
	Zero the adjoints of every element in a vector. More...
template<typename T >
auto	abs (const var_value< T > &a)
	Return the absolute value of the variable (std). More...
var	abs (const std::complex< var > &z)
	Return the absolute value of the complex argument. More...
var	acos (const var &x)
	Return the principal value of the arc cosine of a variable, in radians (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	acos (const VarMat &x)
	Return the principal value of the arc cosine of a variable, in radians (cmath). More...
std::complex< var >	acos (const std::complex< var > &x)
	Return the arc cosine of the complex argument. More...
var	acosh (const var &x)
	The inverse hyperbolic cosine function for variables (C99). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	acosh (const VarMat &x)
	The inverse hyperbolic cosine function for variables (C99). More...
std::complex< var >	acosh (const std::complex< var > &z)
	Return the hyperbolic arc cosine of the complex argument. More...
template<typename T , require_var_t< T > * = nullptr>
auto &	adjoint_of (const T &x)
	Returns a reference to a variable's adjoint. More...
template<typename T , require_not_var_t< T > * = nullptr>
internal::nonexisting_adjoint	adjoint_of (const T &x)
	Returns a reference to a variable's adjoint. More...
template<typename Scal , typename RowVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_row_vector< RowVec >> * = nullptr>
auto	append_col (const Scal &A, const var_value< RowVec > &B)
	Return the result of stacking an scalar on top of the a row vector, with the result being a row vector. More...
template<typename RowVec , typename Scal , require_t< is_eigen_row_vector< RowVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>
auto	append_col (const var_value< RowVec > &A, const Scal &B)
	Return the result of stacking a row vector on top of the an scalar, with the result being a row vector. More...
template<typename Scal , typename ColVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_col_vector< ColVec >> * = nullptr>
auto	append_row (const Scal &A, const var_value< ColVec > &B)
	Return the result of stacking an scalar on top of the a vector, with the result being a vector. More...
template<typename ColVec , typename Scal , require_t< is_eigen_col_vector< ColVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>
auto	append_row (const var_value< ColVec > &A, const Scal &B)
	Return the result of stacking a vector on top of the an scalar, with the result being a vector. More...
var	arg (const std::complex< var > &z)
	Return the phase angle of the complex argument. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	as_array_or_scalar (T &&v)
	Converts a var_value<T> with inner Eigen matrix type to an var_value<T> with an inner array. More...
int	as_bool (const var &v)
	Return 1 if the argument is unequal to zero and 0 otherwise. More...
var	asin (const var &x)
	Return the principal value of the arc sine, in radians, of the specified variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	asin (const VarMat &x)
	Return the principal value of the arc sine, in radians, of the specified variable (cmath). More...
std::complex< var >	asin (const std::complex< var > &z)
	Return the arc sine of the complex argument. More...
var	asinh (const var &x)
	The inverse hyperbolic sine function for variables (C99). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	asinh (const VarMat &x)
	The inverse hyperbolic sine function for variables (C99). More...
std::complex< var >	asinh (const std::complex< var > &z)
	Return the hyperbolic arcsine of the complex argument. More...
var	atan (const var &x)
	Return the principal value of the arc tangent, in radians, of the specified variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	atan (const VarMat &x)
	Return the principal value of the arc tangent, in radians, of the specified variable (cmath). More...
std::complex< var >	atan (const std::complex< var > &z)
	Return the arc tangent of the complex argument. More...
var	atan2 (const var &a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath). More...
var	atan2 (const var &a, double b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath). More...
var	atan2 (double a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath). More...
var	atanh (const var &x)
	The inverse hyperbolic tangent function for variables (C99). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	atanh (const VarMat &x)
	The inverse hyperbolic tangent function for variables (C99). More...
std::complex< var >	atanh (const std::complex< var > &z)
	Return the hyperbolic arc tangent of the complex argument. More...
var	bessel_first_kind (int v, const var &a)
template<typename T1 , typename T2 , require_st_integral< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr>
auto	bessel_first_kind (const T1 &v, const var_value< T2 > &a)
	Overload with var_value<Matrix> for int, std::vector<int>, and std::vector<std::vector<int>> More...
var	bessel_second_kind (int v, const var &a)
template<typename T1 , typename T2 , require_st_integral< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr>
auto	bessel_second_kind (const T1 &v, const var_value< T2 > &a)
	Overload with var_value<Matrix> for int, std::vector<int>, and std::vector<std::vector<int>> More...
var	beta (const var &a, const var &b)
	Returns the beta function and gradients for two var inputs. More...
var	beta (const var &a, double b)
	Returns the beta function and gradient for first var input. More...
var	beta (double a, const var &b)
	Returns the beta function and gradient for second var input. More...
template<typename Mat1 , typename Mat2 , require_any_var_matrix_t< Mat1, Mat2 > * = nullptr, require_all_matrix_t< Mat1, Mat2 > * = nullptr>
auto	beta (const Mat1 &a, const Mat2 &b)
template<typename Scalar , typename VarMat , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr>
auto	beta (const Scalar &a, const VarMat &b)
template<typename VarMat , typename Scalar , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr>
auto	beta (const VarMat &a, const Scalar &b)
var	binary_log_loss (int y, const var &y_hat)
	The log loss function for variables (stan). More...
template<typename Mat , require_eigen_t< Mat > * = nullptr>
auto	binary_log_loss (int y, const var_value< Mat > &y_hat)
	Overload with int and var_value<Matrix> More...
template<typename StdVec , typename Mat , require_eigen_t< Mat > * = nullptr, require_st_integral< StdVec > * = nullptr>
auto	binary_log_loss (const StdVec &y, const var_value< Mat > &y_hat)
	Overload with var_value<Matrix> for std::vector<int> and std::vector<std::vector<int>> More...
var	cbrt (const var &a)
	Returns the cube root of the specified variable (C99). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	cbrt (const VarMat &a)
	Returns the cube root of the specified variable (C99). More...
var	ceil (const var &a)
	Return the ceiling of the specified variable (cmath). More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	ceil (const var_value< T > &a)
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cholesky_corr_constrain (const T &y, int K)
	Return the Cholesky factor of the correlation matrix of the sepcified size read from the unconstrained vector y. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cholesky_corr_constrain (const T &y, int K, scalar_type_t< T > &lp)
	Return the Cholesky factor of the correlation matrix of the sepcified size read from the unconstrained vector y. More...
template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>
auto	cholesky_decompose (const EigMat &A)
	Reverse mode specialization of cholesky decomposition. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	cholesky_decompose (const T &A)
	Reverse mode specialization of Cholesky decomposition. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cholesky_factor_constrain (const T &x, int M, int N)
	Return the Cholesky factor of the specified size read from the specified vector. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cholesky_factor_constrain (const T &x, int M, int N, scalar_type_t< T > &lp)
	Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform. More...
template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>
auto	columns_dot_product (const Mat1 &v1, const Mat2 &v2)
	Returns the dot product of columns of the specified matrices. More...
template<typename Mat , require_eigen_vt< is_var, Mat > * = nullptr>
Eigen::Matrix< var, 1, Mat::ColsAtCompileTime >	columns_dot_self (const Mat &x)
	Returns the dot product of each column of a matrix with itself. More...
template<typename Mat , require_var_matrix_t< Mat > * = nullptr>
auto	columns_dot_self (const Mat &x)
	Returns the dot product of each column of a matrix with itself. More...
std::complex< var >	conj (const std::complex< var > &z)
	Return the complex conjugate of the complex argument. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	corr_matrix_constrain (const T &x, Eigen::Index k)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	corr_matrix_constrain (const T &x, Eigen::Index k, scalar_type_t< T > &lp)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
var	cos (var a)
	Return the cosine of a radian-scaled variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	cos (const VarMat &a)
	Return the cosine of a radian-scaled variable (cmath). More...
std::complex< var >	cos (const std::complex< var > &z)
	Return the cosine of the complex argument. More...
var	cosh (const var &a)
	Return the hyperbolic cosine of the specified variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	cosh (const VarMat &a)
	Return the hyperbolic cosine of the specified variable (cmath). More...
std::complex< var >	cosh (const std::complex< var > &z)
	Return the hyperbolic cosine of the complex argument. More...
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	cov_exp_quad (const std::vector< T_x > &x, const var &sigma, const var &l)
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	cov_exp_quad (const std::vector< T_x > &x, double sigma, const var &l)
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cov_matrix_constrain (const T &x, Eigen::Index K)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cov_matrix_constrain (const T &x, Eigen::Index K, scalar_type_t< T > &lp)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cov_matrix_constrain_lkj (const T &x, size_t k)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values. More...
template<typename T , require_var_vector_t< T > * = nullptr>
var_value< Eigen::MatrixXd >	cov_matrix_constrain_lkj (const T &x, size_t k, scalar_type_t< T > &lp)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant. More...
template<typename T1 , typename T2 , require_any_rev_matrix_t< T1, T2 > * = nullptr>
auto	csr_matrix_times_vector (int m, int n, const T1 &w, const std::vector< int > &v, const std::vector< int > &u, const T2 &b)
template<typename T , require_rev_matrix_t< T > * = nullptr>
var	determinant (const T &m)
var	digamma (const var &a)
	Return the derivative of the log gamma function at the specified value. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	digamma (const T &a)
	Return the elementwise derivative of the log gamma function at the given input vector. More...
template<typename T >
void	dims (const var_value< T > &x, std::vector< int > &result)
	Pushes dimensions of given argument into given result vector. More...
template<typename T >
void	dims (const vari_value< T > &x, std::vector< int > &result)
	Pushes dimensions of given argument into given result vector. More...
template<typename T1 , typename T2 , require_all_vector_t< T1, T2 > * = nullptr, require_not_complex_t< return_type_t< T1, T2 >> * = nullptr, require_all_not_std_vector_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>
var	dot_product (const T1 &v1, const T2 &v2)
	Returns the dot product. More...
template<typename T , require_eigen_vector_vt< is_var, T > * = nullptr>
var	dot_self (const T &v)
	Returns the dot product of a vector of var with itself. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	eigenvalues_sym (const T &m)
	Return the eigenvalues of the specified symmetric matrix. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	eigenvectors_sym (const T &m)
	Return the eigenvectors of the specified symmetric matrix. More...
var	erf (const var &a)
	The error function for variables (C99). More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	erf (const var_value< T > &a)
var	erfc (const var &a)
	The complementary error function for variables (C99). More...
template<typename T , require_matrix_t< T > * = nullptr>
auto	erfc (const var_value< T > &a)
var	exp (const var &a)
	Return the exponentiation of the specified variable (cmath). More...
std::complex< var >	exp (const std::complex< var > &z)
	Return the exponentiation (base e) of the specified complex number. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	exp (const T &x)
	Return the exponentiation of the elements of x. More...
var	exp2 (const var &a)
	Exponentiation base 2 function for variables (C99). More...
template<typename T , require_eigen_t< T > * = nullptr>
auto	exp2 (const var_value< T > &a)
var	expm1 (const var &a)
	The exponentiation of the specified variable minus 1 (C99). More...
template<typename T , require_eigen_t< T > * = nullptr>
auto	expm1 (const var_value< T > &a)
var	fabs (const var &a)
	Return the absolute value of the variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	fabs (const VarMat &a)
	Return the absolute value of the variable (cmath). More...
var	falling_factorial (const var &a, int b)
template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_st_integral< T2 > * = nullptr>
auto	falling_factorial (const var_value< T1 > &a, const T2 &b)
var	fdim (const var &a, const var &b)
	Return the positive difference between the first variable's the value and the second's (C99, C++11). More...
var	fdim (double a, const var &b)
	Return the positive difference between the first value and the value of the second variable (C99, C++11). More...
var	fdim (const var &a, double b)
	Return the positive difference between the first variable's value and the second value (C99, C++11). More...
template<typename VarMat , typename S , require_var_matrix_t< VarMat > * = nullptr, require_var_t< S > * = nullptr>
void	fill (VarMat &x, const S &y)
	Fill the specified container with the specified value. More...
var	floor (const var &a)
	Return the floor of the specified variable (cmath). More...
template<typename T , require_eigen_t< T > * = nullptr>
auto	floor (const var_value< T > &a)
var	fma (const var &x, const var &y, const var &z)
	The fused multiply-add function for three variables (C99). More...
template<typename Tc , require_arithmetic_t< Tc > * = nullptr>
var	fma (const var &x, const var &y, Tc &&z)
	The fused multiply-add function for two variables and a value (C99). More...
template<typename Ta , typename Tb , typename Tc , require_arithmetic_t< Tb > * = nullptr, require_all_var_t< Ta, Tc > * = nullptr>
var	fma (Ta &&x, Tb &&y, Tc &&z)
	The fused multiply-add function for a variable, value, and variable (C99). More...
template<typename Tb , typename Tc , require_all_arithmetic_t< Tb, Tc > * = nullptr>
var	fma (const var &x, Tb &&y, Tc &&z)
	The fused multiply-add function for a variable and two values (C99). More...
template<typename Ta , typename Tc , require_all_arithmetic_t< Ta, Tc > * = nullptr>
var	fma (Ta &&x, const var &y, Tc &&z)
	The fused multiply-add function for a value, variable, and value (C99). More...
template<typename Ta , typename Tb , require_all_arithmetic_t< Ta, Tb > * = nullptr>
var	fma (Ta &&x, Tb &&y, const var &z)
	The fused multiply-add function for two values and a variable, and value (C99). More...
template<typename Ta , require_arithmetic_t< Ta > * = nullptr>
var	fma (Ta &&x, const var &y, const var &z)
	The fused multiply-add function for a value and two variables (C99). More...
template<typename T1 , typename T2 , typename T3 , require_any_matrix_t< T1, T2, T3 > * = nullptr, require_var_t< return_type_t< T1, T2, T3 >> * = nullptr>
auto	fma (const T1 &x, const T2 &y, const T3 &z)
	The fused multiply-add function for three variables (C99). More...
var	fmax (const var &a, const var &b)
	Returns the maximum of the two variable arguments (C99). More...
var	fmax (const var &a, double b)
	Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
var	fmax (double a, const var &b)
	Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
var	fmin (const var &a, const var &b)
	Returns the minimum of the two variable arguments (C99). More...
var	fmin (const var &a, double b)
	Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
var	fmin (double a, const var &b)
	Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
var	fmod (const var &a, const var &b)
	Return the floating point remainder after dividing the first variable by the second (cmath). More...
var	fmod (const var &a, double b)
	Return the floating point remainder after dividing the the first variable by the second scalar (cmath). More...
var	fmod (double a, const var &b)
	Return the floating point remainder after dividing the first scalar by the second variable (cmath). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
Eigen::Matrix< var, T::RowsAtCompileTime, T::ColsAtCompileTime >	from_var_value (const T &a)
	Converts var_value into an Eigen Matrix. More...
template<typename T , require_any_t< conjunction< is_eigen< T >, is_var< scalar_type_t< T >>>, std::is_same< std::decay_t< T >, var >, bool_constant<!std::is_same< scalar_type_t< T >, var >::value >> * = nullptr>
T	from_var_value (T &&a)
	This is a no-op for Eigen containers of vars, scalars or prim types. More...
template<typename T >
auto	from_var_value (const std::vector< T > &a)
	Convert the elements of the std::vector input to var_value types if possible. More...
var	gamma_p (const var &a, const var &b)
var	gamma_p (const var &a, double b)
var	gamma_p (double a, const var &b)
var	gamma_q (const var &a, const var &b)
var	gamma_q (const var &a, double b)
var	gamma_q (double a, const var &b)
template<typename VarMat , require_rev_matrix_t< VarMat > * = nullptr>
auto	generalized_inverse (const VarMat &G)
template<typename T_x , typename T_sigma , require_st_arithmetic< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
Eigen::Matrix< var,-1,-1 >	gp_exp_quad_cov (const std::vector< T_x > &x, const T_sigma sigma, const var length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename T_sigma , require_st_arithmetic< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic >	gp_periodic_cov (const std::vector< T_x > &x, const T_sigma sigma, const var l, const var p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \). More...
void	grad (var &v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &x, Eigen::VectorXd &g)
	Propagate chain rule to calculate gradients starting from the specified variable. More...
void	grad_inc_beta (var &g1, var &g2, const var &a, const var &b, const var &z)
	Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments. More...
var	hypot (const var &a, const var &b)
	Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99). More...
var	hypot (const var &a, double b)
	Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99). More...
var	hypot (double a, const var &b)
	Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99). More...
var	if_else (bool c, const var &y_true, const var &y_false)
	If the specified condition is true, return the first variable, otherwise return the second variable. More...
var	if_else (bool c, double y_true, const var &y_false)
	If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable. More...
var	if_else (bool c, const var &y_true, double y_false)
	If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar. More...
var	inc_beta (const var &a, const var &b, const var &c)
template<typename VarMat , typename S , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>
void	initialize_fill (VarMat &x, const S &y)
	Fill the specified container with the specified value. More...
void	initialize_variable (var &variable, const var &value)
	Initialize variable to value. More...
template<int R, int C>
void	initialize_variable (Eigen::Matrix< var, R, C > &matrix, const var &value)
	Initialize every cell in the matrix to the specified value. More...
template<typename T >
void	initialize_variable (std::vector< T > &variables, const var &value)
	Initialize the variables in the standard vector recursively. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	inv (const var_value< T > &a)
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	inv_cloglog (const var_value< T > &a)
	Return the inverse complementary log-log function applied specified variable (stan). More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	inv_logit (const var_value< T > &a)
	The inverse logit function for variables (stan). More...
var	inv_Phi (const var &p)
	The inverse of unit normal cumulative density function. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	inv_Phi (const T &p)
	Return the elementwise inverse of unit normal cumulative density function. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	inv_sqrt (const var_value< T > &a)
var	inv_square (const var &a)
	\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	inverse (const T &m)
	Reverse mode specialization of calculating the inverse of the matrix. More...
int	is_inf (const var &v)
	Returns 1 if the input's value is infinite and 0 otherwise. More...
bool	is_nan (const var &v)
	Returns 1 if the input's value is NaN and 0 otherwise. More...
bool	is_uninitialized (var x)
	Returns true if the specified variable is uninitialized. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	lambert_w0 (const var_value< T > &a)
	Return the Lambert W function on W0 branch applied to the specified variable. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	lambert_wm1 (const var_value< T > &a)
	Return the Lambert W function on W-1 branch applied to the specified variable. More...
template<typename T , typename L , require_all_stan_scalar_t< T, L > * = nullptr, require_any_var_t< T, L > * = nullptr>
auto	lb_constrain (const T &x, const L &lb, var &lp)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T , typename L , require_matrix_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_any_st_var< T, L > * = nullptr>
auto	lb_constrain (const T &x, const L &lb)
	Specialization of lb_constrain to apply a scalar lower bound elementwise to each input. More...
var	lbeta (const var &a, const var &b)
	Returns the natural logarithm of the beta function and its gradients. More...
var	lbeta (const var &a, double b)
	Returns the natural logarithm of the beta function and its gradients. More...
var	lbeta (double a, const var &b)
	Returns the natural logarithm of the beta function and its gradients. More...
var	ldexp (const var &a, int b)
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	lgamma (const var_value< T > &a)
	The log gamma function for variables (C99). More...
var	lmgamma (int a, const var &b)
var	lmultiply (const var &a, const var &b)
	Return the value of a*log(b). More...
var	lmultiply (const var &a, double b)
	Return the value of a*log(b). More...
var	lmultiply (double a, const var &b)
	Return the value of a*log(b). More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log (const var_value< T > &a)
	Return the natural log of the specified variable (cmath). More...
std::complex< var >	log (const std::complex< var > &z)
	Return the natural logarithm (base e) of the specified complex argument. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log10 (const var_value< T > &a)
	Return the base 10 log of the specified variable (cmath). More...
std::complex< var >	log10 (const std::complex< var > &z)
	Return the base 10 logarithm of the specified complex number. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log1m (const var_value< T > &a)
	The log (1 - x) function for variables. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log1m_exp (const var_value< T > &x)
	Return the log of 1 minus the exponential of the specified variable. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log1m_inv_logit (const var_value< T > &u)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log1p (const var_value< T > &a)
	The log (1 + x) function for variables (C99). More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log1p_exp (const var_value< T > &a)
	Return the log of 1 plus the exponential of the specified variable. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	log2 (const var_value< T > &a)
	Returns the base 2 logarithm of the specified variable (C99). More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
var	log_determinant (const T &m)
template<typename T , require_rev_matrix_t< T > * = nullptr>
var	log_determinant_ldlt (LDLT_factor< T > &A)
	Returns the log det of the matrix whose LDLT factorization is given. More...
template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>
var	log_determinant_spd (const EigMat &m)
	Returns the log det of a symmetric, positive-definite matrix. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
var	log_determinant_spd (const T &m)
	Returns the log det of a symmetric, positive-definite matrix. More...
var	log_diff_exp (const var &a, const var &b)
	Returns the log difference of the exponentiated arguments. More...
var	log_diff_exp (const var &a, double b)
	Returns the log difference of the exponentiated arguments. More...
var	log_diff_exp (double a, const var &b)
	Returns the log difference of the exponentiated arguments. More...
var	log_falling_factorial (const var &a, double b)
var	log_falling_factorial (const var &a, const var &b)
var	log_falling_factorial (double a, const var &b)
var	log_inv_logit (const var &u)
	Return the natural logarithm of the inverse logit of the specified argument. More...
var	log_inv_logit_diff (const var &a, double b)
var	log_inv_logit_diff (const var &a, const var &b)
var	log_inv_logit_diff (double a, const var &b)
void	log_mix_partial_helper (double theta_val, double lambda1_val, double lambda2_val, double &one_m_exp_lam2_m_lam1, double &one_m_t_prod_exp_lam2_m_lam1, double &one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1)
template<typename T_theta , typename T_lambda1 , typename T_lambda2 , require_any_var_t< T_theta, T_lambda1, T_lambda2 > * = nullptr>
return_type_t< T_theta, T_lambda1, T_lambda2 >	log_mix (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2)
	Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
var	log_rising_factorial (const var &a, double b)
var	log_rising_factorial (const var &a, const var &b)
var	log_rising_factorial (double a, const var &b)
var	log_sum_exp (const var &a, const var &b)
	Returns the log sum of exponentials. More...
var	log_sum_exp (const var &a, double b)
	Returns the log sum of exponentials. More...
var	log_sum_exp (double a, const var &b)
	Returns the log sum of exponentials. More...
template<typename T , require_eigen_st< is_var, T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>
var	log_sum_exp (const T &v)
	Returns the log sum of exponentials of the input. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
var	log_sum_exp (const T &x)
	Returns the log sum of exponentials of the input. More...
template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>
auto	logit (const var_value< T > &u)
	Return the log odds of the specified argument. More...
template<typename T , typename L , typename U , require_all_matrix_t< T, L > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_var_t< return_type_t< T, L, U >> * = nullptr>
auto	lub_constrain (const T &x, const L &lb, const U &ub, std::decay_t< return_type_t< T, L, U >> &lp)
	Specialization for Eigen matrix with matrix lower bound and scalar upper bound plus lp. More...
template<typename Ta , typename Tb , require_all_eigen_t< Ta, Tb > * = nullptr, require_any_st_autodiff< Ta, Tb > * = nullptr>
Eigen::Matrix< return_type_t< Ta, Tb >,-1, Tb::ColsAtCompileTime >	matrix_exp_multiply (const Ta &A, const Tb &B)
	Wrapper of matrix_exp_action function for a more literal name. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
plain_type_t< T >	matrix_power (const T &M, const int n)
	Returns the nth power of the specific matrix. More...
template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>
auto	mdivide_left (const T1 &A, const T2 &B)
	Return the solution X of AX = B. More...
template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>
auto	mdivide_left_ldlt (LDLT_factor< T1 > &A, const T2 &B)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_matrix_base_vt< is_var, EigMat1, EigMat2 > * = nullptr>
Eigen::Matrix< var, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_left_spd (const EigMat1 &A, const EigMat2 &b)
template<typename EigMat1 , typename EigMat2 , require_eigen_matrix_base_vt< is_var, EigMat1 > * = nullptr, require_eigen_matrix_base_vt< std::is_arithmetic, EigMat2 > * = nullptr>
Eigen::Matrix< var, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime >	mdivide_left_spd (const EigMat1 &A, const EigMat2 &b)
template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_var_matrix_t< T1, T2 > * = nullptr>
auto	mdivide_left_spd (const T1 &A, const T2 &B)
	Returns the solution of the system Ax=B where A is symmetric positive definite. More...
template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_eigen_vt< is_var, T1, T2 > * = nullptr>
Eigen::Matrix< var, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri (const T1 &A, const T2 &b)
template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_eigen_vt< std::is_arithmetic, T1 > * = nullptr, require_eigen_vt< is_var, T2 > * = nullptr>
Eigen::Matrix< var, T1::RowsAtCompileTime, T2::ColsAtCompileTime >	mdivide_left_tri (const T1 &A, const T2 &b)
template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_var_matrix_t< T1, T2 > * = nullptr>
auto	mdivide_left_tri (const T1 &A, const T2 &B)
	Returns the solution of the system Ax=B when A is triangular. More...
var	modified_bessel_first_kind (int v, const var &a)
var	modified_bessel_second_kind (int v, const var &a)
template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_return_type_t< is_var, T1, T2 > * = nullptr, require_row_and_col_vector_t< T1, T2 > * = nullptr>
var	multiply (const T1 &A, const T2 &B)
	Return the product of a row vector times a column vector as a scalar. More...
template<typename T1 , typename T2 , require_not_matrix_t< T1 > * = nullptr, require_matrix_t< T2 > * = nullptr, require_return_type_t< is_var, T1, T2 > * = nullptr, require_not_row_and_col_vector_t< T1, T2 > * = nullptr>
auto	multiply (const T1 &A, const T2 &B)
	Return specified matrix multiplied by specified scalar where at least one input has a scalar type of a var_value. More...
var	multiply_log (const var &a, const var &b)
	Return the value of a*log(b). More...
var	multiply_log (const var &a, double b)
	Return the value of a*log(b). More...
var	multiply_log (double a, const var &b)
	Return the value of a*log(b). More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	multiply_lower_tri_self_transpose (const T &L)
var	norm (const std::complex< var > &z)
	Return the squared magnitude of the complex argument. More...
template<typename T , require_rev_col_vector_t< T > * = nullptr>
auto	ordered_constrain (const T &x)
	Return an increasing ordered vector derived from the specified free vector. More...
template<typename VarVec , require_var_col_vector_t< VarVec > * = nullptr>
auto	ordered_constrain (const VarVec &x, scalar_type_t< VarVec > &lp)
	Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
var	owens_t (const var &h, const var &a)
	The Owen's T function of h and a. More...
var	owens_t (const var &h, double a)
	The Owen's T function of h and a. More...
var	owens_t (double h, const var &a)
	The Owen's T function of h and a. More...
var	Phi (const var &a)
	The unit normal cumulative density function for variables (stan). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	Phi (const T &a)
	Elementwise unit normal cumulative density function for varmat types. More...
var	Phi_approx (const var &a)
	Approximation of the unit normal CDF for variables (stan). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	Phi_approx (const T &a)
std::complex< var >	polar (const var &r, const var &theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T >
std::complex< var >	polar (T r, const var &theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T >
std::complex< var >	polar (const var &r, T theta)
	Returns complex number with specified magnitude and phase angle. More...
template<typename T , require_rev_col_vector_t< T > * = nullptr>
auto	positive_ordered_constrain (const T &x)
	Return an increasing positive ordered vector derived from the specified free vector. More...
var	pow (const var &base, const var &exponent)
	Return the base raised to the power of the exponent (cmath). More...
template<typename T , typename = require_arithmetic_t<T>>
var	pow (const var &base, T exponent)
	Return the base variable raised to the power of the exponent scalar (cmath). More...
template<typename T , typename = require_arithmetic_t<T>>
var	pow (T base, const var &exponent)
	Return the base scalar raised to the power of the exponent variable (cmath). More...
std::complex< var >	pow (const std::complex< var > &x, const std::complex< var > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (const std::complex< var > &x, const std::complex< T > y)
	Return the first argument raised to the power of the second argument. More...
std::complex< var >	pow (const std::complex< var > &x, const var &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (const std::complex< var > &x, T y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (std::complex< T > x, const std::complex< var > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (std::complex< T > x, const var &y)
	Return the first argument raised to the power of the second argument. More...
std::complex< var >	pow (const var &x, const std::complex< var > &y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (const var &x, std::complex< T > y)
	Return the first argument raised to the power of the second argument. More...
template<typename T , typename = require_arithmetic_t<T>>
std::complex< var >	pow (T x, const std::complex< var > &y)
	Return the first argument raised to the power of the second argument. More...
std::complex< var >	pow (const std::complex< var > &x, int y)
	Return the first argument raised to the power of the second argument. More...
double	primitive_value (const var &v)
	Return the primitive double value for the specified autodiff variable. More...
std::complex< var >	proj (const std::complex< var > &z)
	Return the projection of the complex argument onto the Riemann sphere. More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_var< EigMat1, EigMat2 > * = nullptr>
promote_scalar_t< var, EigMat2 >	quad_form (const EigMat1 &A, const EigMat2 &B, bool symmetric=false)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_var< EigMat, ColVec > * = nullptr>
var	quad_form (const EigMat &A, const ColVec &B, bool symmetric=false)
	Return the quadratic form \( B^T A B \). More...
template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_not_col_vector_t< Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>
auto	quad_form (const Mat1 &A, const Mat2 &B, bool symmetric=false)
	Return the quadratic form \( B^T A B \). More...
template<typename Mat , typename Vec , require_matrix_t< Mat > * = nullptr, require_col_vector_t< Vec > * = nullptr, require_any_var_matrix_t< Mat, Vec > * = nullptr>
var	quad_form (const Mat &A, const Vec &B, bool symmetric=false)
	Return the quadratic form \( B^T A B \). More...
template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_vt_var< EigMat1, EigMat2 > * = nullptr>
auto	quad_form_sym (const EigMat1 &A, const EigMat2 &B)
	Return the quadratic form \( B^T A B \) of a symmetric matrix. More...
template<typename T , require_var_vector_t< T > * = nullptr>
auto	read_corr_L (const T &CPCs, size_t K)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T1 , typename T2 , require_var_vector_t< T1 > * = nullptr, require_stan_scalar_t< T2 > * = nullptr>
auto	read_corr_L (const T1 &CPCs, size_t K, T2 &log_prob)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T_CPCs , require_var_vector_t< T_CPCs > * = nullptr>
var_value< Eigen::MatrixXd >	read_corr_matrix (const T_CPCs &CPCs, size_t K)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T_CPCs , require_var_vector_t< T_CPCs > * = nullptr>
var_value< Eigen::MatrixXd >	read_corr_matrix (const T_CPCs &CPCs, size_t K, scalar_type_t< T_CPCs > &log_prob)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T_CPCs , typename T_sds , require_any_var_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>
auto	read_cov_L (const T_CPCs &CPCs, const T_sds &sds, scalar_type_t< T_CPCs > &log_prob)
	This is the function that should be called prior to evaluating the density of any elliptical distribution. More...
template<typename T_CPCs , typename T_sds , require_all_var_vector_t< T_CPCs, T_sds > * = nullptr>
var_value< Eigen::MatrixXd >	read_cov_matrix (const T_CPCs &CPCs, const T_sds &sds, scalar_type_t< T_CPCs > &log_prob)
	A generally worse alternative to call prior to evaluating the density of an elliptical distribution. More...
template<typename T_CPCs , typename T_sds , require_all_var_vector_t< T_CPCs, T_sds > * = nullptr>
var_value< Eigen::MatrixXd >	read_cov_matrix (const T_CPCs &CPCs, const T_sds &sds)
	Builds a covariance matrix from CPCs and standard deviations. More...
template<typename T_ret , require_var_matrix_t< T_ret > * = nullptr, require_eigen_row_vector_t< value_type_t< T_ret >> * = nullptr>
auto	rep_row_vector (var x, int n)
	Overload for var_value<Vector>. More...
template<typename T_ret , require_var_matrix_t< T_ret > * = nullptr, require_eigen_col_vector_t< value_type_t< T_ret >> * = nullptr>
auto	rep_vector (var x, int n)
	Overload for var_value<Vector>. More...
var	rising_factorial (const var &a, int b)
var	round (const var &a)
	Returns the rounded form of the specified variable (C99). More...
template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_any_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>
Eigen::Matrix< var, Mat1::RowsAtCompileTime, 1 >	rows_dot_product (const Mat1 &v1, const Mat2 &v2)
	Returns the dot product of rows of the specified matrices. More...
template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>
auto	rows_dot_product (const Mat1 &v1, const Mat2 &v2)
	Returns the dot product of rows of the specified matrices. More...
template<typename Mat , require_eigen_vt< is_var, Mat > * = nullptr>
Eigen::Matrix< var, Mat::RowsAtCompileTime, 1 >	rows_dot_self (const Mat &x)
	Returns the dot product of each row of a matrix with itself. More...
template<typename Mat , require_var_matrix_t< Mat > * = nullptr>
auto	rows_dot_self (const Mat &x)
	Returns the dot product of row row of a matrix with itself. More...
template<typename T , require_eigen_st< is_var, T > * = nullptr>
var	sd (const T &x)
	Return the sample standard deviation of a variable which inherits from EigenBase. More...
template<typename T , require_std_vector_st< is_var, T > * = nullptr>
auto	sd (const T &m)
	Return the sample standard deviation of the specified std vector, column vector, row vector, matrix, or std vector of any of these types. More...
template<typename T , require_rev_col_vector_t< T > * = nullptr>
auto	simplex_constrain (const T &y)
	Return the simplex corresponding to the specified free vector. More...
template<typename T , require_rev_col_vector_t< T > * = nullptr>
auto	simplex_constrain (const T &y, scalar_type_t< T > &lp)
	Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
var	sin (const var &a)
	Return the sine of a radian-scaled variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	sin (const VarMat &a)
	Return the sine of a radian-scaled variable (cmath). More...
std::complex< var >	sin (const std::complex< var > &z)
	Return the sine of the complex argument. More...
template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>
auto	singular_values (const EigMat &m)
	Return the singular values of the specified matrix. More...
var	sinh (const var &a)
	Return the hyperbolic sine of the specified variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	sinh (const VarMat &a)
	Return the hyperbolic of a radian-scaled variable (cmath). More...
std::complex< var >	sinh (const std::complex< var > &z)
	Return the hyperbolic sine of the complex argument. More...
template<typename Mat , require_rev_matrix_t< Mat > * = nullptr>
auto	softmax (const Mat &alpha)
	Return the softmax of the specified Eigen vector. More...
var	sqrt (const var &a)
	Return the square root of the specified variable (cmath). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	sqrt (const T &a)
	Return elementwise square root of vector. More...
std::complex< var >	sqrt (const std::complex< var > &z)
	Return the square root of the complex argument. More...
var	square (const var &x)
	Return the square of the input variable. More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	square (const T &x)
	Return the elementwise square of x. More...
var	squared_distance (const var &a, const var &b)
	Returns the squared distance. More...
var	squared_distance (const var &a, double b)
	Returns the squared distance. More...
var	squared_distance (double a, const var &b)
	Returns the squared distance. More...
template<typename EigVecVar1 , typename EigVecVar2 , require_all_eigen_vector_vt< is_var, EigVecVar1, EigVecVar2 > * = nullptr>
var	squared_distance (const EigVecVar1 &v1, const EigVecVar2 &v2)
template<typename EigVecVar , typename EigVecArith , require_eigen_vector_vt< is_var, EigVecVar > * = nullptr, require_eigen_vector_vt< std::is_arithmetic, EigVecArith > * = nullptr>
var	squared_distance (const EigVecVar &v1, const EigVecArith &v2)
template<typename EigVecArith , typename EigVecVar , require_eigen_vector_vt< std::is_arithmetic, EigVecArith > * = nullptr, require_eigen_vector_vt< is_var, EigVecVar > * = nullptr>
var	squared_distance (const EigVecArith &v1, const EigVecVar &v2)
template<typename T1 , typename T2 , require_all_vector_t< T1, T2 > * = nullptr, require_any_var_vector_t< T1, T2 > * = nullptr>
var	squared_distance (const T1 &A, const T2 &B)
	Compute the squared distance between the elements in two inputs. More...
void	stan_print (std::ostream *o, const var &x)
var	step (const var &a)
	Return the step, or heaviside, function applied to the specified variable (stan). More...
template<typename Alloc >
var	sum (const std::vector< var, Alloc > &m)
	Returns the sum of the entries of the specified vector. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
var	sum (const T &x)
	Returns the sum of the coefficients of the specified matrix. More...
template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>
auto	svd_U (const EigMat &m)
	Given input matrix m, return matrix U where m = UDV^{T} More...
template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>
auto	svd_V (const EigMat &m)
	Given input matrix m, return matrix V where m = UDV^{T} More...
var	tan (const var &a)
	Return the tangent of a radian-scaled variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	tan (const VarMat &a)
	Return the tangent of a radian-scaled variable (cmath). More...
std::complex< var >	tan (const std::complex< var > &z)
	Return the tangent of the complex argument. More...
var	tanh (const var &a)
	Return the hyperbolic tangent of the specified variable (cmath). More...
template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>
auto	tanh (const VarMat &a)
	Return the hyperbolic tangent of elements of a. More...
std::complex< var >	tanh (const std::complex< var > &z)
	Return the hyperbolic tangent of the complex argument. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	tcrossprod (const T &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...
var	tgamma (const var &a)
	Return the Gamma function applied to the specified variable (C99). More...
template<typename T , require_var_matrix_t< T > * = nullptr>
auto	tgamma (const T &a)
	Return elementwise gamma function. More...
template<typename T , require_not_same_t< T, arena_t< T >> * = nullptr, require_not_container_t< T > * = nullptr, require_not_matrix_cl_t< T > * = nullptr>
arena_t< T >	to_arena (T &&a)
	Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered. More...
template<typename T , require_same_t< T, arena_t< T >> * = nullptr, require_not_matrix_cl_t< T > * = nullptr, require_not_std_vector_t< T > * = nullptr>
std::remove_reference_t< T >	to_arena (T &&a)
	Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered. More...
template<typename T , require_eigen_t< T > * = nullptr, require_not_same_t< T, arena_t< T >> * = nullptr>
arena_t< T >	to_arena (const T &a)
	Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered. More...
template<typename T >
std::vector< T, arena_allocator< T > >	to_arena (const std::vector< T, arena_allocator< T >> &a)
	Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered. More...
template<typename T , require_same_t< T, arena_t< T >> * = nullptr>
arena_t< std::vector< T > >	to_arena (const std::vector< T > &a)
	Converts given argument into a type that has any dynamic allocation on AD stack. More...
template<bool Condition, typename T , std::enable_if_t<!Condition > * = nullptr>
T	to_arena_if (T &&a)
	If the condition is true, converts given argument into a type that has any dynamic allocation on AD stack. More...
template<bool Condition, typename T , std::enable_if_t< Condition > * = nullptr>
arena_t< T >	to_arena_if (const T &a)
var	to_var (double x)
	Converts argument to an automatic differentiation variable. More...
var &	to_var (var &x)
	Specialization of to_var for non-const var input. More...
const var &	to_var (const var &x)
	Specialization of to_var for const var input. More...
std::vector< var >	to_var (const std::vector< double > &v)
	Converts argument to an automatic differentiation variable. More...
const std::vector< var > &	to_var (const std::vector< var > &v)
	Specialization of to_var to for const input vector of var. More...
std::vector< var > &	to_var (std::vector< var > &v)
	Specialization of to_var to for non-const input vector of var. More...
matrix_v	to_var (const matrix_d &m)
	Converts argument to an automatic differentiation variable. More...
matrix_v &	to_var (matrix_v &m)
	Specialization of to_var for non-const matrices of vars. More...
const matrix_v &	to_var (const matrix_v &m)
	Specialization of to_var for const matrices of vars. More...
vector_v	to_var (const vector_d &v)
	Converts argument to an automatic differentiation variable. More...
const vector_v &	to_var (const vector_v &v)
	Specialization of to_var for const column vector of vars. More...
vector_v &	to_var (vector_v &v)
	Specialization of to_var for non-const column vector of vars. More...
row_vector_v	to_var (const row_vector_d &rv)
	Converts argument to an automatic differentiation variable. More...
const row_vector_v &	to_var (const row_vector_v &rv)
	Specialization of to_var for const row vector of vars. More...
row_vector_v &	to_var (row_vector_v &rv)
	Specialization of to_var for non-const row vector of vars. More...
template<typename T , require_eigen_vt< is_var, T > * = nullptr>
var_value< Eigen::Matrix< double, T::RowsAtCompileTime, T::ColsAtCompileTime > >	to_var_value (const T &a)
	Converts an Eigen matrix (or vector or row_vector) or expression of vars into var_value. More...
template<typename T , require_var_t< T > * = nullptr>
T	to_var_value (T &&a)
	This is a no-op for var_values. More...
template<typename T >
auto	to_var_value (const std::vector< T > &a)
	Convert the elements of the std::vector input to var_value types if possible. More...
template<typename EigMat , require_eigen_t< EigMat > * = nullptr>
auto	to_vector (const var_value< EigMat > &x)
	Reshape a var_value<Matrix> to a var_value<ColumnVector>. More...
template<typename T , require_rev_matrix_t< T > * = nullptr>
auto	trace (const T &m)
	Returns the trace of the specified matrix. More...
template<typename Td , typename Ta , typename Tb , require_not_col_vector_t< Td > * = nullptr, require_all_matrix_t< Td, Ta, Tb > * = nullptr, require_any_st_var< Td, Ta, Tb > * = nullptr>
var	trace_gen_inv_quad_form_ldlt (const Td &D, LDLT_factor< Ta > &A, const Tb &B)
	Compute the trace of an inverse quadratic form premultiplied by a square matrix. More...
template<typename Td , typename Ta , typename Tb , require_col_vector_t< Td > * = nullptr, require_all_matrix_t< Ta, Tb > * = nullptr, require_any_st_var< Td, Ta, Tb > * = nullptr>
var	trace_gen_inv_quad_form_ldlt (const Td &D, const LDLT_factor< Ta > &A, const Tb &B)
	Compute the trace of an inverse quadratic form. More...
template<typename Td , typename Ta , typename Tb , typename = require_any_var_t<value_type_t<Td>, value_type_t<Ta>, value_type_t<Tb>>, typename = require_all_eigen_t<Td, Ta, Tb>>
var	trace_gen_quad_form (const Td &D, const Ta &A, const Tb &B)
	Return the trace of D times the quadratic form of B and A. More...
template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>
var	trace_inv_quad_form_ldlt (LDLT_factor< T1 > &A, const T2 &B)
	Compute the trace of an inverse quadratic form premultiplied by a square matrix. More...
template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>
var	trace_quad_form (const Mat1 &A, const Mat2 &B)
	Compute trace(B^T A B). More...
var	trigamma (const var &u)
	Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument). More...
var	trunc (const var &a)
	Returns the truncation of the specified variable (C99). More...
template<typename T , typename U , require_matrix_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_any_st_var< T, U > * = nullptr>
auto	ub_constrain (const T &x, const U &ub, return_type_t< T, U > &lp)
	Specialization of ub_constrain to apply a scalar upper bound elementwise to each input. More...
template<typename T , require_rev_col_vector_t< T > * = nullptr>
auto	unit_vector_constrain (const T &y)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T , require_eigen_col_vector_vt< is_var, T > * = nullptr>
auto	unit_vector_constrain (const T &y, var &lp)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T >
auto &	value_of (const var_value< T > &v)
	Return the value of the specified variable. More...
template<typename T >
auto	value_of_rec (const var_value< T > &v)
	Return the value of the specified variable. More...
var	variance (const std::vector< var > &v)
	Return the sample variance of the specified standard vector. More...
template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>
var	variance (const EigMat &m)
	Return the sample variance of the specified vector, row vector, or matrix. More...
template<typename Mat , require_var_matrix_t< Mat > * = nullptr>
var	variance (const Mat &x)
	Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero. More...
template<typename F , typename T1 , typename T2 , typename T_u , typename T_f >
Eigen::Matrix< T2,-1, 1 >	algebra_solver_fp (const F &f, const Eigen::Matrix< T1,-1, 1 > &x, const Eigen::Matrix< T2,-1, 1 > &y, const std::vector< double > &x_r, const std::vector< int > &x_i, const std::vector< T_u > &u_scale, const std::vector< T_f > &f_scale, std::ostream *msgs=nullptr, double f_tol=1e-8, int max_num_steps=200)
	Return a fixed pointer to the specified system of algebraic equations of form. More...
template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr, require_all_st_arithmetic< Args... > * = nullptr>
Eigen::VectorXd	algebra_solver_newton_impl (const F &f, const T &x, std::ostream *const msgs, const double scaling_step_size, const double function_tolerance, const int64_t max_num_steps, const Args &...args)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T , typename... T_Args, require_eigen_vector_t< T > * = nullptr, require_any_st_var< T_Args... > * = nullptr>
Eigen::Matrix< var, Eigen::Dynamic, 1 >	algebra_solver_newton_impl (const F &f, const T &x, std::ostream *const msgs, const double scaling_step_size, const double function_tolerance, const int64_t max_num_steps, const T_Args &...args)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>
Eigen::Matrix< scalar_type_t< T2 >, Eigen::Dynamic, 1 >	algebra_solver_newton (const F &f, const T1 &x, const T2 &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *const msgs=nullptr, const double scaling_step_size=1e-3, const double function_tolerance=1e-6, const long int max_num_steps=200)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr>
T &	algebra_solver_powell_call_solver (const F &f, T &x, std::ostream *const msgs, const double relative_tolerance, const double function_tolerance, const int64_t max_num_steps, const Args &...args)
	Solve algebraic equations using Powell solver. More...
template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr, require_all_st_arithmetic< Args... > * = nullptr>
Eigen::VectorXd	algebra_solver_powell_impl (const F &f, const T &x, std::ostream *const msgs, const double relative_tolerance, const double function_tolerance, const int64_t max_num_steps, const Args &...args)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T2 >, Eigen::Dynamic, 1 >	algebra_solver_powell (const F &f, const T1 &x, const T2 &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *const msgs=nullptr, const double relative_tolerance=1e-10, const double function_tolerance=1e-6, const int64_t max_num_steps=1e+3)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>
Eigen::Matrix< value_type_t< T2 >, Eigen::Dynamic, 1 >	algebra_solver (const F &f, const T1 &x, const T2 &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, const double relative_tolerance=1e-10, const double function_tolerance=1e-6, const int64_t max_num_steps=1e+3)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T , typename... T_Args, require_eigen_vector_t< T > * = nullptr, require_any_st_var< T_Args... > * = nullptr>
Eigen::Matrix< var, Eigen::Dynamic, 1 >	algebra_solver_powell_impl (const F &f, const T &x, std::ostream *const msgs, const double relative_tolerance, const double function_tolerance, const int64_t max_num_steps, const T_Args &...args)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename T1 , typename T2 >
void	algebra_solver_check (const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > y, const std::vector< double > &dat, const std::vector< int > &dat_int, double function_tolerance, long int max_num_steps)
void	cvodes_err_handler (int error_code, const char *module, const char *function, char *msg, void *eh_data)
void	cvodes_set_options (void *cvodes_mem, long int max_num_steps)
template<typename F >
void	gradient (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument. More...
template<typename F , typename T_a , typename T_b , typename... Args, require_any_st_var< T_a, T_b, Args... > * = nullptr>
return_type_t< T_a, T_b, Args... >	integrate_1d_impl (const F &f, const T_a &a, const T_b &b, double relative_tolerance, std::ostream *msgs, const Args &...args)
	Return the integral of f from a to b to the given relative tolerance. More...
template<typename F , typename T_a , typename T_b , typename T_theta , typename = require_any_var_t<T_a, T_b, T_theta>>
return_type_t< T_a, T_b, T_theta >	integrate_1d (const F &f, const T_a &a, const T_b &b, const std::vector< T_theta > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, std::ostream *msgs, const double relative_tolerance=std::sqrt(EPSILON))
	Compute the integral of the single variable function f from a to b to within a specified relative tolerance. More...
template<typename F , typename Tpar >
std::vector< std::vector< Tpar > >	integrate_dae (const F &f, const std::vector< double > &yy0, const std::vector< double > &yp0, double t0, const std::vector< double > &ts, const std::vector< Tpar > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, const double rtol, const double atol, const int64_t max_num_steps=idas_integrator::IDAS_MAX_STEPS, std::ostream *msgs=nullptr)
	Return the solutions for a semi-explicit DAE system with residual specified by functor F, given the specified consistent initial state yy0 and yp0. More...
template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >
std::vector< std::vector< return_type_t< T_y0, T_param, T_t0, T_ts > > >	integrate_ode_adams (const F &f, const std::vector< T_y0 > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T_param > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double absolute_tolerance=1e-10, long int max_num_steps=1e8)
template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >
std::vector< std::vector< return_type_t< T_y0, T_param, T_t0, T_ts > > >	integrate_ode_bdf (const F &f, const std::vector< T_y0 > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T_param > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double absolute_tolerance=1e-10, long int max_num_steps=1e8)
template<typename F >
void	jacobian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, Eigen::Matrix< double, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &J)
template<typename F1 , typename... Args>
Eigen::VectorXd	kinsol_solve (const F1 &f, const Eigen::VectorXd &x, const double scaling_step_tol, const double function_tolerance, const int64_t max_num_steps, const bool custom_jacobian, const int steps_eval_jacobian, const int global_line_search, std::ostream *const msgs, const Args &...args)
	Return the solution to the specified algebraic system, given an initial guess. More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_adams_tol_impl (const char *function_name, const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_adams_tol (const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_adams (const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr, require_any_not_st_arithmetic< T_y0, T_t0, T_ts, T_Args... > * = nullptr>
auto	ode_adjoint_impl (const char *function_name, F &&f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance_forward, const T_abs_tol_fwd &absolute_tolerance_forward, double relative_tolerance_backward, const T_abs_tol_bwd &absolute_tolerance_backward, double relative_tolerance_quadrature, double absolute_tolerance_quadrature, long int max_num_steps, long int num_steps_between_checkpoints, int interpolation_polynomial, int solver_forward, int solver_backward, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr, require_all_st_arithmetic< T_y0, T_t0, T_ts, T_Args... > * = nullptr>
std::vector< Eigen::VectorXd >	ode_adjoint_impl (const char *function_name, F &&f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance_forward, const T_abs_tol_fwd &absolute_tolerance_forward, double relative_tolerance_backward, const T_abs_tol_bwd &absolute_tolerance_backward, double relative_tolerance_quadrature, double absolute_tolerance_quadrature, long int max_num_steps, long int num_steps_between_checkpoints, int interpolation_polynomial, int solver_forward, int solver_backward, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr>
auto	ode_adjoint_tol_ctl (F &&f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance_forward, const T_abs_tol_fwd &absolute_tolerance_forward, double relative_tolerance_backward, const T_abs_tol_bwd &absolute_tolerance_backward, double relative_tolerance_quadrature, double absolute_tolerance_quadrature, long int max_num_steps, long int num_steps_between_checkpoints, int interpolation_polynomial, int solver_forward, int solver_backward, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_bdf_tol_impl (const char *function_name, const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_bdf_tol (const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, double relative_tolerance, double absolute_tolerance, long int max_num_steps, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>
std::vector< Eigen::Matrix< stan::return_type_t< T_y0, T_t0, T_ts, T_Args... >, Eigen::Dynamic, 1 > >	ode_bdf (const F &f, const T_y0 &y0, const T_t0 &t0, const std::vector< T_ts > &ts, std::ostream *msgs, const T_Args &...args)
	Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ... More...
template<typename F , typename T_y0_t0 , typename T_t0 , typename T_t , typename... Args, require_any_autodiff_t< T_y0_t0, T_t0, T_t, scalar_type_t< Args >... > * = nullptr>
Eigen::Matrix< var, Eigen::Dynamic, 1 >	ode_store_sensitivities (const F &f, const std::vector< double > &coupled_state, const Eigen::Matrix< T_y0_t0, Eigen::Dynamic, 1 > &y0, const T_t0 &t0, const T_t &t, std::ostream *msgs, const Args &...args)
	Build output vars for a state of the ODE solve, storing the sensitivities precomputed using the forward sensitivity problem in precomputed varis. More...
template<typename T , typename L , typename U , require_not_std_vector_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr>
auto	lub_free (T &&y, L &&lb, U &&ub)
	lub_free Return the unconstrained variable that transforms to the y given the specified bounds. More...
template<typename T , typename L , typename U , require_all_eigen_t< T, L, U > * = nullptr>
auto	lub_free (T &&y, L &&lb, U &&ub)
	Overload for matrix constrained variable, matrix upper bound, matrix lower bound. More...
template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>
auto	lub_free (const std::vector< T > y, const L &lb, const U &ub)
	Overload for std::vector constrained variable. More...
template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>
auto	lub_free (const std::vector< T > y, const L &lb, const std::vector< U > &ub)
	Overload for std::vector constrained variable and std::vector upper bound. More...
template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>
auto	lub_free (const std::vector< T > y, const std::vector< L > &lb, const U &ub)
	Overload for std::vector constrained variable and std::vector lower bound. More...
template<typename T , typename L , typename U >
auto	lub_free (const std::vector< T > y, const std::vector< L > &lb, const std::vector< U > &ub)
	Overload for std::vector constrained variable and std::vector constraints. More...

Variables
static opencl_context	opencl_context
const double	CONSTRAINT_TOLERANCE = 1E-8
	The tolerance for checking arithmetic bounds in rank and in simplexes. More...
static constexpr double	EPSILON = std::numeric_limits<double>::epsilon()
	Smallest positive value. More...
static constexpr double	INFTY = std::numeric_limits<double>::infinity()
	Positive infinity. More...
static constexpr double	NEGATIVE_INFTY = -INFTY
	Negative infinity. More...
static constexpr double	NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()
	(Quiet) not-a-number value. More...
static constexpr double	TWO_PI = boost::math::constants::two_pi<double>()
	Twice the value of \( \pi \), \( 2\pi \). More...
static constexpr double	LOG_ZERO = -INFTY
	The natural logarithm of 0, \( \log 0 \). More...
const double	LOG_EPSILON = std::log(EPSILON)
	The natural logarithm of machine precision \( \epsilon \), \( \log \epsilon \). More...
const double	LOG_PI = std::log(pi())
	The natural logarithm of \( \pi \), \( \log \pi \). More...
static constexpr double	LOG_TWO = boost::math::constants::ln_two<double>()
	The natural logarithm of 2, \( \log 2 \). More...
static constexpr double	LOG_HALF = -LOG_TWO
	The natural logarithm of 0.5, \( \log 0.5 = \log 1 - \log 2 \). More...
const double	LOG_TWO_PI = LOG_TWO + LOG_PI
	The natural logarithm of 2 plus the natural logarithm of \( \pi \), \( \log(2\pi) \). More...
const double	LOG_PI_OVER_FOUR = 0.25 * LOG_PI
	The value of one quarter the natural logarithm of \( \pi \), \( \log(\pi) / 4 \). More...
const double	LOG_SQRT_PI = std::log(boost::math::constants::root_pi<double>())
	The natural logarithm of the square root of \( \pi \), \( \log(sqrt{\pi}) \). More...
static constexpr double	LOG_TEN = boost::math::constants::ln_ten<double>()
	The natural logarithm of 10, \( \log 10 \). More...
static constexpr double	SQRT_TWO = boost::math::constants::root_two<double>()
	The value of the square root of 2, \( \sqrt{2} \). More...
static constexpr double	SQRT_PI = boost::math::constants::root_pi<double>()
	The value of the square root of \( \pi \), \( \sqrt{\pi} \). More...
static constexpr double	SQRT_TWO_PI = boost::math::constants::root_two_pi<double>()
	The value of the square root of \( 2\pi \), \( \sqrt{2\pi} \). More...
static constexpr double	SQRT_TWO_OVER_SQRT_PI = SQRT_TWO / SQRT_PI
	The square root of 2 divided by the square root of \( \pi \), \( \sqrt{2} / \sqrt{\pi} \). More...
static constexpr double	INV_SQRT_TWO = boost::math::constants::one_div_root_two<double>()
	The value of 1 over the square root of 2, \( 1 / \sqrt{2} \). More...
static constexpr double	INV_SQRT_PI = boost::math::constants::one_div_root_pi<double>()
	The value of 1 over the square root of \( \pi \), \( 1 / \sqrt{\pi} \). More...
static constexpr double	INV_SQRT_TWO_PI = boost::math::constants::one_div_root_two_pi<double>()
	The value of 1 over the square root of \( 2\pi \), \( 1 / \sqrt{2\pi} \). More...
static constexpr double	TWO_OVER_SQRT_PI = boost::math::constants::two_div_root_pi<double>()
	The value of 2 over the square root of \( \pi \), \( 2 / \sqrt{\pi} \). More...
static constexpr double	HALF_LOG_TWO = 0.5 * LOG_TWO
	The value of half the natural logarithm 2, \( \log(2) / 2 \). More...
const double	HALF_LOG_TWO_PI = 0.5 * LOG_TWO_PI
	The value of half the natural logarithm \( 2\pi \), \( \log(2\pi) / 2 \). More...
const double	NEG_LOG_SQRT_TWO_PI = -std::log(SQRT_TWO_PI)
	The value of minus the natural logarithm of the square root of \( 2\pi \), \( -\log(\sqrt{2\pi}) \). More...
const double	POISSON_MAX_RATE = std::pow(2.0, 30)
	Largest rate parameter allowed in Poisson RNG. More...
constexpr double	lgamma_stirling_diff_useful = 10
const std::string	MAJOR_VERSION = "4"
	Major version number for Stan math library. More...
const std::string	MINOR_VERSION = "2"
	Minor version number for Stan math library. More...
const std::string	PATCH_VERSION = "1"
	Patch version for Stan math library. More...

---

Class Documentation

struct stan::math::apply_scalar_unary

struct stan::math::apply_vector_unary

struct stan::math::coupled_ode_system_impl

struct stan::math::FixedPointSolver

class stan::math::LDLT_factor

struct stan::math::pass_type

Class Members
typedef const T &	type

struct stan::math::pass_type< double >

Class Members
typedef double	type

struct stan::math::pass_type< int >

Class Members
typedef int	type

struct stan::math::promote_scalar_type

Class Members
typedef T	type	The promoted type.

struct stan::math::promote_scalar_type< T, S, require_all_t< is_var< T >, is_var< S >, is_eigen< value_type_t< S > > > >

Class Members
typedef decay_t< S >	type	The promoted type.

struct stan::math::promote_scalar_type< T, S, require_all_t< std::is_arithmetic< T >, is_var< S >, is_eigen< value_type_t< S > > > >

Class Members
typedef typename promote_scalar_type< T, value_type_t< S >>::type	type	The promoted type.

struct stan::math::promote_scalar_type< T, S, require_eigen_t< S > >

Class Members
typedef typename conditional < is_same< typename traits < decay_t< S >>::XprKind, MatrixXpr >::value, Matrix < typename promote_scalar_type < T, typename Scalar >::type, RowsAtCompileTime, ColsAtCompileTime >, Array < typename promote_scalar_type < T, typename Scalar >::type, RowsAtCompileTime, ColsAtCompileTime >>::type	type	The promoted type.

struct stan::math::promote_scalar_type< T, std::vector< S > >

Class Members
typedef vector< typename promote_scalar_type< T, S > ::type >	type	The promoted type.

struct stan::math::ref_type_for_opencl

Class Members
typedef conditional_t < is_rvalue_reference< T > ::value, T_val, const T & >	T_optionally_ref
typedef pinned_matrix < T_plain_col_major >	T_pinned
typedef conditional_t< is_same < typename traits< T_val > ::XprKind, MatrixXpr >::value, Matrix< value_type_t< T > , RowsAtCompileTime, ColsAtCompileTime >, Array < value_type_t< T > , RowsAtCompileTime, ColsAtCompileTime >>	T_plain_col_major
typedef remove_reference_t< T >	T_val
typedef decay_t< decltype(declval < T_val >()	T_val_derived
typedef conditional_t< traits < Ref< decay_t < T_plain_col_major > >>::template match < T_val_derived > ::MatchAtCompileTime &&(evaluator< T_val_derived > ::Flags &LinearAccessBit)&&(evaluator < T_val_derived >::Flags &PacketAccessBit), T_optionally_ref, T_pinned >	type

struct stan::math::ref_type_for_opencl< T, require_arena_matrix_t< T > >

Class Members
typedef typename ref_type_for_opencl< typename decay_t< T >::Base >::type	type

struct stan::math::ref_type_for_opencl< T, require_not_eigen_t< T > >

Class Members
typedef conditional_t < is_rvalue_reference< T > ::value, remove_reference_t< T > , const T & >	type

struct stan::math::store_type

Class Members
typedef const T &	type

struct stan::math::store_type< double >

Class Members
typedef const double	type

struct stan::math::store_type< int >

Class Members
typedef const int	type

class stan::math::var_value

class stan::math::vari_view

Typedef Documentation

template<int B = 0>

using stan::math::boost_policy_t = typedef boost::math::policies::policy< boost::math::policies::overflow_error< boost::math::policies::errno_on_error>, boost::math::policies::pole_error<boost::math::policies::errno_on_error>, boost::math::policies::promote_double<false>, boost::math::policies::digits2<B>>

Boost policy that overrides the defaults to match the built-in C++ standard library functions.

The non-default behavior from Boost's built-ins are (1) overflow errors return error numbers on error. (2) pole errors return error numbers on error. (3) doubles passed to Boost are not promoted to long double.

The policy is equipped with an optional generic argument B controlling the precision in some functions. If set to 0, the maximum precision available in the type being used is demanded from Boost. Otherwise, it correspond to the approximately B-bit precision, i.e. for trading speed for accuracy.

Definition at line 30 of file boost_policy.hpp.

using stan::math::ChainableStack = typedef AutodiffStackSingleton<vari_base, chainable_alloc>

Definition at line 9 of file chainablestack.hpp.

template<typename... Types>

using stan::math::common_scalar_t = typedef typename std::common_type_t< typename std::remove_reference_t<Types>::Scalar...>

Wrapper for std::common_type_t

Definition at line 13 of file common_return_scalar.hpp.

template<typename T >

typedef stan::math::index_type_t< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >> stan::math::size_type

Type for sizes and indexes in an Eigen matrix with double elements.

Definition at line 38 of file index_type.hpp.

using stan::math::matrix_d = typedef Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>

Type for matrix of double values.

Definition at line 19 of file typedefs.hpp.

using stan::math::matrix_fd = typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 13 of file typedefs.hpp.

using stan::math::matrix_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 16 of file typedefs.hpp.

using stan::math::matrix_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 15 of file typedefs.hpp.

using stan::math::matrix_fv = typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 12 of file typedefs.hpp.

using stan::math::matrix_v = typedef Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic>

The type of a matrix holding var values.

Definition at line 18 of file typedefs.hpp.

using stan::math::matrix_vi = typedef Eigen::Matrix<vari*, Eigen::Dynamic, Eigen::Dynamic>

The type of a matrix holding vari* values.

Definition at line 36 of file typedefs.hpp.

using stan::math::precomputed_gradients_vari = typedef precomputed_gradients_vari_template<std::tuple<>, std::tuple<>>

Definition at line 189 of file precomputed_gradients.hpp.

using stan::math::profile_key = typedef std::pair<std::string, std::thread::id>

Definition at line 116 of file profiling.hpp.

using stan::math::profile_map = typedef std::map<profile_key, profile_info>

Definition at line 118 of file profiling.hpp.

template<typename T , typename S >

using stan::math::promote_scalar_t = typedef typename promote_scalar_type<std::decay_t<T>, std::decay_t<S>>::type

Definition at line 98 of file promote_scalar_type.hpp.

template<typename T >

using stan::math::ref_type_for_opencl_t = typedef typename ref_type_for_opencl<T>::type

Definition at line 64 of file ref_type_for_opencl.hpp.

using stan::math::row_vector_d = typedef Eigen::Matrix<double, 1, Eigen::Dynamic>

Type for (row) vector of double values.

Definition at line 29 of file typedefs.hpp.

using stan::math::row_vector_fd = typedef Eigen::Matrix<fvar<double>, 1, Eigen::Dynamic>

Definition at line 22 of file typedefs.hpp.

using stan::math::row_vector_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, 1, Eigen::Dynamic>

Definition at line 24 of file typedefs.hpp.

using stan::math::row_vector_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, 1, Eigen::Dynamic>

Definition at line 23 of file typedefs.hpp.

using stan::math::row_vector_fv = typedef Eigen::Matrix<fvar<var>, 1, Eigen::Dynamic>

Definition at line 21 of file typedefs.hpp.

using stan::math::row_vector_v = typedef Eigen::Matrix<var, 1, Eigen::Dynamic>

The type of a row vector holding var values.

Definition at line 30 of file typedefs.hpp.

using stan::math::row_vector_vi = typedef Eigen::Matrix<vari*, 1, Eigen::Dynamic>

The type of a row vector holding vari* values.

Definition at line 48 of file typedefs.hpp.

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index stan::math::size_type

Type for sizes and indexes in an Eigen matrix with double elements.

Definition at line 11 of file typedefs.hpp.

using stan::math::var = typedef var_value<double>

Definition at line 1065 of file var.hpp.

using stan::math::vari = typedef vari_value<double>

Definition at line 197 of file vari.hpp.

using stan::math::vector_d = typedef Eigen::Matrix<double, Eigen::Dynamic, 1>

Type for (column) vector of double values.

Definition at line 24 of file typedefs.hpp.

using stan::math::vector_fd = typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, 1>

Definition at line 18 of file typedefs.hpp.

using stan::math::vector_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, 1>

Definition at line 20 of file typedefs.hpp.

using stan::math::vector_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, Eigen::Dynamic, 1>

Definition at line 19 of file typedefs.hpp.

using stan::math::vector_fv = typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, 1>

Definition at line 17 of file typedefs.hpp.

using stan::math::vector_v = typedef Eigen::Matrix<var, Eigen::Dynamic, 1>

The type of a (column) vector holding var values.

Definition at line 24 of file typedefs.hpp.

using stan::math::vector_vi = typedef Eigen::Matrix<vari*, Eigen::Dynamic, 1>

The type of a (column) vector holding vari* values.

Definition at line 42 of file typedefs.hpp.

Enumeration Type Documentation

enum stan::math::matrix_cl_view

strong

Enumerator
Diagonal
Lower
Upper
Entire

Definition at line 11 of file matrix_cl_view.hpp.

Function Documentation

template<typename T >

fvar<T> stan::math::abs ( const fvar< T > &  x )

inline

Definition at line 15 of file abs.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

T stan::math::abs

(

T

x

)

Return the absolute value of the specified arithmetic argument.

The return type is the same as the argument type.

Template Parameters

T	type of arument (must be arithmetic)

Parameters

x

argument

Returns

absolute value of argument

Definition at line 25 of file abs.hpp.

template<typename T >

fvar<T> stan::math::abs ( const std::complex< fvar< T >> &  z )

inline

Return the absolute value of the complex argument.

Template Parameters

T	value type of argument

Parameters

[in]

z

argument

Returns

absolute value of the argument

Definition at line 35 of file abs.hpp.

template<typename T >

std::vector<T> stan::math::abs

(

const std::vector< T > &

x

)

Definition at line 37 of file abs.hpp.

template<typename T >

auto stan::math::abs ( const var_value< T > &  a )

inline

Return the absolute value of the variable (std).

Delegates to fabs() (see for doc).

\[ \mbox{abs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{abs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	A floating point type or an Eigen type with floating point scalar.

Parameters

a	Variable input.

Returns

Absolute value of variable.

Definition at line 41 of file abs.hpp.

var stan::math::abs ( const std::complex< var > &  z )

inline

Return the absolute value of the complex argument.

Parameters

[in]

z

argument

Returns

absolute value of the argument

Definition at line 51 of file abs.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::abs

(

const Eigen::Matrix< T, R, C > &

x

)

Return the elementwise absolute value of the specified matrix, vector, or row vector.

Template Parameters

T	type of scalar for matrix argument (real or complex)
R	row specification (1 or -1)
C	column specification (1 or -1)

Parameters

x

argument

Returns

elementwise absolute value of argument

Definition at line 55 of file abs.hpp.

template<typename T , require_complex_t< T > * = nullptr>

auto stan::math::abs

(

T

x

)

Return the absolute value (also known as the norm, modulus, or magnitude) of the specified complex argument.

Template Parameters

T	type of argument (must be complex)

Parameters

x

argument

Returns

absolute value of argument (a real number)

Definition at line 68 of file abs.hpp.

template<typename... Pargs>

double * stan::math::accumulate_adjoints ( double *  dest, const var &  x, Pargs &&...  args  )

inline

Accumulate adjoints from x into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer.

Template Parameters

Pargs

Types of remaining arguments

Parameters

dest	Pointer to where adjoints are to be accumulated
x	A var
args	Further args to accumulate over

Returns

Final position of adjoint storage pointer

Definition at line 51 of file accumulate_adjoints.hpp.

template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr, typename... Pargs>

double * stan::math::accumulate_adjoints ( double *  dest, VarVec &&  x, Pargs &&...  args  )

inline

Accumulate adjoints from std::vector x into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer.

Template Parameters

Pargs

Types of remaining arguments

Parameters

dest	Pointer to where adjoints are to be accumulated
x	A std::vector of vars
args	Further args to accumulate over

Returns

Final position of adjoint storage pointer

Definition at line 71 of file accumulate_adjoints.hpp.

template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr, typename... Pargs>

double * stan::math::accumulate_adjoints ( double *  dest, VecContainer &&  x, Pargs &&...  args  )

inline

Accumulate adjoints from x (a std::vector of containers containing vars) into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer.

Template Parameters

VecContainer	the type of a standard container holding var containers.
Pargs	Types of remaining arguments

Parameters

dest	Pointer to where adjoints are to be accumulated
x	A std::vector of containers holding vars
args	Further args to accumulate over

Returns

Final position of adjoint storage pointer

Definition at line 96 of file accumulate_adjoints.hpp.

template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr, typename... Pargs>

double * stan::math::accumulate_adjoints ( double *  dest, EigT &&  x, Pargs &&...  args  )

inline

Accumulate adjoints from x (an Eigen type containing vars) into storage pointed to by dest, increment the adjoint storage pointer, recursively accumulate the adjoints of the rest of the arguments, and return final position of storage pointer.

Template Parameters

EigT	Type derived from EigenBase containing vars.
Pargs	Types of remaining arguments

Parameters

dest	Pointer to where adjoints are to be accumulated
x	An eigen type holding vars to accumulate over
args	Further args to accumulate over

Returns

Final position of adjoint storage pointer

Definition at line 119 of file accumulate_adjoints.hpp.

template<typename Arith , require_st_arithmetic< Arith > * = nullptr, typename... Pargs>

double * stan::math::accumulate_adjoints ( double *  dest, Arith &&  x, Pargs &&...  args  )

inline

Ignore arithmetic types.

Recursively accumulate the adjoints of the rest of the arguments and return final position of adjoint storage pointer.

Template Parameters

Arith	A type satisfying std::is_arithmetic.
Pargs	Types of remaining arguments

Parameters

dest	Pointer to where adjoints are to be accumulated
x	An object that is either arithmetic or a container of Arithmetic types
args	Further args to accumulate over

Returns

Final position of adjoint storage pointer

Definition at line 139 of file accumulate_adjoints.hpp.

double * stan::math::accumulate_adjoints ( double *  dest )

inline

End accumulate_adjoints recursion and return pointer.

Parameters

dest

Pointer

Definition at line 148 of file accumulate_adjoints.hpp.

template<typename T >

fvar<T> stan::math::acos ( const fvar< T > &  x )

inline

Definition at line 15 of file acos.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::acos ( const var_value< T > &  A )

inline

Returns the elementwise acos() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise acos() of the input, in radians.

Definition at line 21 of file acos.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::acos ( const std::complex< fvar< T >> &  x )

inline

Return the arc cosine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

x

argument

Returns

arc cosine of the argument

Definition at line 29 of file acos.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::acos ( const Container &  x )

inline

Returns the elementwise acos() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

argument

Returns

Arc cosine of each variable in the container, in radians.

Definition at line 49 of file acos.hpp.

var stan::math::acos ( const var &  x )

inline

Return the principal value of the arc cosine of a variable, in radians (cmath).

The derivative is defined by

\(\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}\).

\[ \mbox{acos}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arccos(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acos}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arccos(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arccos(x)}{\partial x} = -\frac{1}{\sqrt{1-x^2}} \]

Parameters

x

argument

Returns

Arc cosine of variable, in radians.

Definition at line 55 of file acos.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::acos ( const Container &  x )

inline

Version of acos() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

argument

Returns

Arc cosine of each variable in the container, in radians.

Definition at line 63 of file acos.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::acos ( const VarMat &  x )

inline

Return the principal value of the arc cosine of a variable, in radians (cmath).

Parameters

x	a var_value with inner Eigen type

Returns

Arc cosine of variable, in radians.

Definition at line 69 of file acos.hpp.

std::complex<var> stan::math::acos ( const std::complex< var > &  x )

inline

Return the arc cosine of the complex argument.

Parameters

x

argument

Returns

arc cosine of the argument

Definition at line 83 of file acos.hpp.

template<typename T >

fvar<T> stan::math::acosh ( const fvar< T > &  x )

inline

Definition at line 16 of file acosh.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::acosh ( const var_value< T > &  A )

inline

Returns the elementwise acosh() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise acosh() of the input, in radians.

Definition at line 21 of file acosh.hpp.

double stan::math::acosh ( double  x )

inline

Return the inverse hyperbolic cosine of the specified value.

Returns nan for nan argument.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic cosine of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 26 of file acosh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::acosh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic arc cosine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic arc cosine of the argument

Definition at line 29 of file acosh.hpp.

double stan::math::acosh ( int  x )

inline

Integer version of acosh.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic cosine of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 46 of file acosh.hpp.

var stan::math::acosh ( const var &  x )

inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see acosh().

The derivative is defined by

\(\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}\).

\[ \mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \]

\[ \frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}} \]

Parameters

x	The variable.

Returns

Inverse hyperbolic cosine of the variable.

Definition at line 63 of file acosh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::acosh ( const VarMat &  x )

inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see acosh().

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

x	The variable

Returns

Inverse hyperbolic cosine of the variable.

Definition at line 78 of file acosh.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::acosh ( const T &  x )

inline

Return the elementwise application of acosh() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise acosh of members of container.

Definition at line 89 of file acosh.hpp.

std::complex<var> stan::math::acosh ( const std::complex< var > &  z )

inline

Return the hyperbolic arc cosine of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic arc cosine of the argument

Definition at line 93 of file acosh.hpp.

template<typename ScalarA , typename ScalarB , require_all_stan_scalar_t< ScalarA, ScalarB > * = nullptr, require_all_not_var_t< ScalarA, ScalarB > * = nullptr>

return_type_t<ScalarA, ScalarB> stan::math::add ( const ScalarA &  a, const ScalarB &  b  )

inline

Return the sum of the specified scalars.

Template Parameters

ScalarA	type of the first scalar
ScalarB	type of the second scalar

Parameters

a	first scalar
b	second scalar

Returns

the sum of the scalars

Definition at line 23 of file add.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::add ( T_a &&  a, T_b &&  b  )

inline

Addition of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

The sum of the given arguments

Definition at line 31 of file add.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>

auto stan::math::add ( const Mat1 &  m1, const Mat2 &  m2  )

inline

Return the sum of the specified matrices.

The two matrices must have the same dimensions.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	First matrix or expression.
m2	Second matrix or expression.

Returns

Sum of the matrices.

Exceptions

std::invalid_argument

if m1 and m2 do not have the same dimensions.

Definition at line 43 of file add.hpp.

template<typename Mat , typename Scal , require_eigen_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr, require_all_not_st_var< Mat, Scal > * = nullptr>

auto stan::math::add ( const Mat &  m, const Scal  c  )

inline

Return the sum of the specified matrix and specified scalar.

Template Parameters

Mat	type of the matrix or expression
Scal	type of the scalar

Parameters

m	Matrix or expression.
c	Scalar.

Returns

The matrix plus the scalar.

Definition at line 60 of file add.hpp.

template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>

auto stan::math::add ( const Scal  c, const Mat &  m  )

inline

Return the sum of the specified scalar and specified matrix.

Template Parameters

Scal	type of the scalar
Mat	type of the matrix or expression

Parameters

c	Scalar.
m	Matrix.

Returns

The scalar plus the matrix.

Definition at line 76 of file add.hpp.

template<typename VarMat1 , typename VarMat2 , require_all_rev_matrix_t< VarMat1, VarMat2 > * = nullptr>

auto stan::math::add ( const VarMat1 &  a, const VarMat2 &  b  )

inline

Addition operator for matrix variables (C++).

Template Parameters

VarMat1	A matrix of vars or a var with an underlying matrix type.
VarMat2	A matrix of vars or a var with an underlying matrix type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 111 of file operator_addition.hpp.

template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::add ( const VarMat &  a, const Arith &  b  )

inline

Addition operator for a matrix variable and arithmetic (C++).

Template Parameters

VarMat	A matrix of vars or a var with an underlying matrix type.
Arith	A type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 142 of file operator_addition.hpp.

template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::add ( const Arith &  a, const VarMat &  b  )

inline

Addition operator for an arithmetic type and matrix variable (C++).

Template Parameters

VarMat	A matrix of vars or a var with an underlying matrix type.
Arith	A type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 168 of file operator_addition.hpp.

template<typename Var , typename EigMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr>

auto stan::math::add ( const Var &  a, const EigMat &  b  )

inline

Addition operator for an arithmetic matrix and variable (C++).

Template Parameters

Var	A var_value with an underlying arithmetic type.
EigMat	An Eigen Matrix type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 184 of file operator_addition.hpp.

template<typename EigMat , typename Var , require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>

auto stan::math::add ( const EigMat &  a, const Var &  b  )

inline

Addition operator for a variable and arithmetic matrix (C++).

Template Parameters

EigMat	An Eigen Matrix type with an arithmetic Scalar type.
Var	A var_value with an underlying arithmetic type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 203 of file operator_addition.hpp.

template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::add ( const Var &  a, const VarMat &  b  )

inline

Addition operator for a variable and variable matrix (C++).

Template Parameters

Var	A var_value with an underlying arithmetic type.
VarMat	An Eigen Matrix type with a variable Scalar type or a var_value with an underlying matrix type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 220 of file operator_addition.hpp.

template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::add ( const VarMat &  a, const Var &  b  )

inline

Addition operator for a variable matrix and variable (C++).

Template Parameters

VarMat	An Eigen Matrix type with a variable Scalar type or a var_value with an underlying matrix type.
Var	A var_value with an underlying arithmetic type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 249 of file operator_addition.hpp.

template<typename T1 , typename T2 , require_any_var_vt< std::is_arithmetic, T1, T2 > * = nullptr, require_any_arithmetic_t< T1, T2 > * = nullptr>

auto stan::math::add ( const T1 &  a, const T2 &  b  )

inline

Definition at line 256 of file operator_addition.hpp.

template<typename T1 , typename T2 , require_all_var_vt< std::is_arithmetic, T1, T2 > * = nullptr>

auto stan::math::add ( const T1 &  a, const T2 &  b  )

inline

Definition at line 262 of file operator_addition.hpp.

template<typename T_m , typename T_a , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr, require_all_kernel_expressions_t< T_a > * = nullptr>

auto stan::math::add_diag ( T_m &&  mat, T_a &&  to_add  )

inline

Returns a Matrix with values added along the main diagonal.

Template Parameters

T_m	type of input kernel generator expression for the input matrix
T_a	type of input kernel generator expression to add along the diagonal

Parameters

mat	input kernel generator expression
to_add	scalar value or input kernel generator expression to add along the diagonal

Returns

a kernel generator expressio with to_add added along main diagonal

Definition at line 27 of file add_diag.hpp.

template<typename T_m , typename T_a , typename = require_eigen_t<T_m>, typename = require_any_t<is_eigen_vector<T_a>, is_stan_scalar<T_a>>>

Eigen::Matrix<return_type_t<T_m, T_a>, Eigen::Dynamic, Eigen::Dynamic> stan::math::add_diag ( const T_m &  mat, const T_a &  to_add  )

inline

Returns a Matrix with values added along the main diagonal.

Template Parameters

T_m	type of element in Eigen::Matrix
T_a	Type of element(s) to add along the diagonal

Parameters

mat	a matrix
to_add	scalar value or column vector or row vector to add along the diagonal

Returns

a matrix with to_add added along main diagonal

Definition at line 29 of file add_diag.hpp.

template<typename T_m , typename T_a , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_m > * = nullptr, require_all_prim_or_rev_kernel_expression_t< T_a > * = nullptr, require_any_var_t< T_m, T_a > * = nullptr>

auto stan::math::add_diag ( const T_m &  mat, const T_a &  to_add  )

inline

Returns a Matrix with values added along the main diagonal.

Template Parameters

T_m	type of input kernel generator expression for the input matrix
T_a	type of input kernel generator expression to add along the diagonal

Parameters

mat	input kernel generator expression
to_add	scalar value or input kernel generator expression to add along the diagonal

Returns

a kernel generator expressio with to_add added along main diagonal

Definition at line 31 of file add_diag.hpp.

template<typename T , require_var_t< T > * = nullptr>

auto& stan::math::adjoint_of

(

const T &

x

)

Returns a reference to a variable's adjoint.

Parameters

x

a var

Returns

reference to x's adjoint

Definition at line 37 of file adjoint_of.hpp.

template<typename T , require_not_var_t< T > * = nullptr>

internal::nonexisting_adjoint stan::math::adjoint_of

(

const T &

x

)

Returns a reference to a variable's adjoint.

If the input object is not var, it does not have an adjoint and this returns a dummy object. It defines operators += and -=, but they should not actually be called.

Parameters

x	any non-var object

Returns

a dummy adjoint

Definition at line 50 of file adjoint_of.hpp.

template<typename... T_results>

adjoint_results_cl<T_results...> stan::math::adjoint_results

(

T_results &&...

results

)

Deduces types for constructing adjoint_results_cl object.

Template Parameters

T_results

types of results

Parameters

results

results that will be calculated.

Definition at line 145 of file adjoint_results.hpp.

template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T2>, Eigen::Dynamic, 1> stan::math::algebra_solver	(	const F &	f,
		const T1 &	x,
		const T2 &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		const double	relative_tolerance = 1e-10,
		const double	function_tolerance = 1e-6,
		const int64_t	max_num_steps = 1e+3
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

Signature to maintain backward compatibility, will be removed in the future.

Template Parameters

F	type of equation system function
T1	type of elements in the x vector
T2	type of elements in the y vector

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in]	y	parameter vector for the equation system.
[in]	dat	continuous data vector for the equation system.
[in]	dat_int	integer data vector for the equation system.
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given y, dat, and dat_int.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::domain_error) if solver exceeds max_num_steps.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::domain_error) if the norm of the solution exceeds the function tolerance.

Definition at line 262 of file algebra_solver_powell.hpp.

template<typename T1 , typename T2 >

void stan::math::algebra_solver_check	(	const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 >	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		double	function_tolerance,
		long int	max_num_steps
	)

Definition at line 101 of file algebra_system.hpp.

template<typename F , typename T1 , typename T2 , typename T_u , typename T_f >

Eigen::Matrix<T2, -1, 1> stan::math::algebra_solver_fp	(	const F &	f,
		const Eigen::Matrix< T1,-1, 1 > &	x,
		const Eigen::Matrix< T2,-1, 1 > &	y,
		const std::vector< double > &	x_r,
		const std::vector< int > &	x_i,
		const std::vector< T_u > &	u_scale,
		const std::vector< T_f > &	f_scale,
		std::ostream *	msgs = nullptr,
		double	f_tol = 1e-8,
		int	max_num_steps = 200
	)

Return a fixed pointer to the specified system of algebraic equations of form.

x = F(x; theta)

given an initial guess x, and parameters theta and data. Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaling controls, the function tolerance, and the maximum number of steps.

Template Parameters

F	type of equation system function.
T	type of initial guess vector. The final solution type doesn't depend on initial guess type, but we allow initial guess to be either data or param.
T_u	type of scaling vector for unknowns. We allow it to be var because scaling could be parameter dependent. Internally these params are converted to data because scaling is applied.
T_f	type of scaling vector for residual. We allow it to be var because scaling could be parameter dependent. Internally these params are converted to data because scaling is applied.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	x_r	Continuous data vector for the equation system.
[in]	x_i	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	u_scale	diagonal scaling matrix elements Du such that Du*x has all components roughly the same magnitude when x is close to a solution. (ref. KINSOL user guide chap.2 sec. "Scaling")
[in]	f_scale	diagonal scaling matrix elements such that Df*(x-f(x)) has all components roughly the same magnitude when x is not too close to a solution. (ref. KINSOL user guide chap.2 sec. "Scaling")
[in]	f_tol	Function-norm stopping tolerance.
[in]	max_num_steps	maximum number of function evaluations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.

Definition at line 365 of file algebra_solver_fp.hpp.

template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>

Eigen::Matrix<scalar_type_t<T2>, Eigen::Dynamic, 1> stan::math::algebra_solver_newton	(	const F &	f,
		const T1 &	x,
		const T2 &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *const	msgs = nullptr,
		const double	scaling_step_size = 1e-3,
		const double	function_tolerance = 1e-6,
		const long int	max_num_steps = 200
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaled step size, the function tolerance, and the maximum number of steps.

Signature to maintain backward compatibility, will be removed in the future.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system.
[in]	dat	Continuous data vector for the equation system.
[in]	dat_int	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_size	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error	if solver exceeds max_num_steps.

Definition at line 248 of file algebra_solver_newton.hpp.

template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr, require_all_st_arithmetic< Args... > * = nullptr>

Eigen::VectorXd stan::math::algebra_solver_newton_impl	(	const F &	f,
		const T &	x,
		std::ostream *const	msgs,
		const double	scaling_step_size,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const Args &...	args
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaled step size, the function tolerance, and the maximum number of steps.

This overload handles non-autodiff parameters.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_size	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in]	args	Additional parameters to the equation system functor.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given args.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error	if solver exceeds max_num_steps.

Definition at line 58 of file algebra_solver_newton.hpp.

template<typename F , typename T , typename... T_Args, require_eigen_vector_t< T > * = nullptr, require_any_st_var< T_Args... > * = nullptr>

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::algebra_solver_newton_impl	(	const F &	f,
		const T &	x,
		std::ostream *const	msgs,
		const double	scaling_step_size,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const T_Args &...	args
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaled step size, the function tolerance, and the maximum number of steps.

This overload handles var parameters.

The Jacobian (J_{xy}) (i.e., Jacobian of unknown (x) w.r.t. the parameter (y)) is calculated given the solution as follows. Since [ f(x, y) = 0, ] we have ((J_{pq}) being the Jacobian matrix ( {dq} {dq})) [

• J_{fx} J_{xy} = J_{fy}, ] and therefore (J_{xy}) can be solved from system [

J_{fx} J_{xy} = J_{fy}. ] Let (eta) be the adjoint with respect to (x); then to calculate [ J_{xy}, ] we solve [

• ( J_{fx}^{-1}) J_{fy}. ]

Template Parameters

F	type of equation system function.
T	type of initial guess vector.
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_size	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in]	args	Additional parameters to the equation system functor.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given args.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error	if solver exceeds max_num_steps.

Definition at line 139 of file algebra_solver_newton.hpp.

template<typename F , typename T1 , typename T2 , require_all_eigen_vector_t< T1, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T2>, Eigen::Dynamic, 1> stan::math::algebra_solver_powell	(	const F &	f,
		const T1 &	x,
		const T2 &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *const	msgs = nullptr,
		const double	relative_tolerance = 1e-10,
		const double	function_tolerance = 1e-6,
		const int64_t	max_num_steps = 1e+3
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

Template Parameters

F	type of equation system function
T1	type of elements in the x vector
T2	type of elements in the y vector

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in]	y	parameter vector for the equation system.
[in]	dat	continuous data vector for the equation system.
[in]	dat_int	integer data vector for the equation system.
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error</code>	solver exceeds max_num_steps.
<code>std::domain_error</code>	if the norm of the solution exceeds the function tolerance.

Definition at line 202 of file algebra_solver_powell.hpp.

template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr>

T& stan::math::algebra_solver_powell_call_solver	(	const F &	f,
		T &	x,
		std::ostream *const	msgs,
		const double	relative_tolerance,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const Args &...	args
	)

Solve algebraic equations using Powell solver.

Template Parameters

F	type of equation system function
T	type of elements in the x vector
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluates the system of equations
[in]	x	Vector of starting values (initial guess).
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in]	args	additional parameters to the equation system functor.

Returns

theta_dbl Double vector of solutions to the system of equations.

Precondition

x has size greater than zero.

x has only finite elements.

f returns finite values when passed any value of x and the given args.

relative_tolerance is non-negative.

function_tolerance is non-negative.

max_num_steps is positive.

Exceptions

<code>std::domain_error</code>	solver exceeds max_num_steps.
<code>std::domain_error</code>	if the norm of the solution exceeds the function tolerance.

Definition at line 48 of file algebra_solver_powell.hpp.

template<typename F , typename T , typename... Args, require_eigen_vector_t< T > * = nullptr, require_all_st_arithmetic< Args... > * = nullptr>

Eigen::VectorXd stan::math::algebra_solver_powell_impl	(	const F &	f,
		const T &	x,
		std::ostream *const	msgs,
		const double	relative_tolerance,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const Args &...	args
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

This overload handles non-autodiff parameters.

Template Parameters

F	type of equation system function
T	type of elements in the x vector
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in]	args	additional parameters to the equation system functor.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given args.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements. elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error</code>	solver exceeds max_num_steps.
<code>std::domain_error</code>	if the norm of the solution exceeds the function tolerance.

Definition at line 129 of file algebra_solver_powell.hpp.

template<typename F , typename T , typename... T_Args, require_eigen_vector_t< T > * = nullptr, require_any_st_var< T_Args... > * = nullptr>

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::algebra_solver_powell_impl	(	const F &	f,
		const T &	x,
		std::ostream *const	msgs,
		const double	relative_tolerance,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const T_Args &...	args
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

This overload handles var parameters.

The Jacobian (J_{xy}) (i.e., Jacobian of unknown (x) w.r.t. the parameter (y)) is calculated given the solution as follows. Since [ f(x, y) = 0, ] we have ((J_{pq}) being the Jacobian matrix ( {dq} {dq})) [

• J_{fx} J_{xy} = J_{fy}, ] and therefore (J_{xy}) can be solved from system [

J_{fx} J_{xy} = J_{fy}. ] Let (eta) be the adjoint with respect to (x); then to calculate [ J_{xy}, ] we solve [

• ( J_{fx}^{-1}) J_{fy}. ]

Template Parameters

F	type of equation system function
T	type of elements in the x vector
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in]	args	Additional parameters to the equation system functor.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given args.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements. elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error</code>	solver exceeds max_num_steps.
<code>std::domain_error</code>	if the norm of the solution exceeds the function tolerance.

Definition at line 335 of file algebra_solver_powell.hpp.

template<typename T_x , typename T_y , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr>

auto stan::math::append_array ( T_x &&  x, T_y &&  y  )

inline

Return the concatenation of two specified vectors in the order of the arguments.

Template Parameters

T1	type of elements in first vector
T2	type of elements in second vector

Parameters

x	First vector
y	Second vector

Returns

A vector of x and y concatenated together (in that order)

Definition at line 24 of file append_array.hpp.

template<typename T1 , typename T2 >

append_return_type<std::vector<T1>, std::vector<T2> >::type stan::math::append_array ( const std::vector< T1 > &  x, const std::vector< T2 > &  y  )

inline

Return the concatenation of two specified vectors in the order of the arguments.

The return type is computed with append_return_type

Template Parameters

T1	type of elements in first vector
T2	type of elements in second vector

Parameters

x	First vector
y	Second vector

Returns

A vector of x and y concatenated together (in that order)

Definition at line 29 of file append_array.hpp.

template<typename T1 >

std::vector<T1> stan::math::append_array ( const std::vector< T1 > &  x, const std::vector< T1 > &  y  )

inline

Return the concatenation of two specified vectors in the order of the arguments.

Template Parameters

T1	type of elements in both vectors

Parameters

x	First vector
y	Second vector

Returns

A vector of x and y concatenated together (in that order)

Definition at line 59 of file append_array.hpp.

template<typename T1 , typename T2 , typename = require_all_eigen_t<T1, T2>>

auto stan::math::append_col ( const T1 &  A, const T2 &  B  )

inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, vector) -> matrix, (vector, matrix) -> matrix, (vector, vector) -> matrix, (row vector, row vector) -> row_vector.

Template Parameters

T1	type of the first matrix
T2	type of the second matrix

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of appending the first matrix followed by the second matrix side by side.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, vector) -> matrix, (vector, matrix) -> matrix, (vector, vector) -> matrix, (row vector, row vector) -> row_vector.

Template Parameters

T1	A var_value with inner matrix type.
T1	A var_value with inner matrix type.

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of appending the first matrix followed by the second matrix side by side.

Definition at line 33 of file append_col.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::append_col ( T_a &&  a, T_b &&  b  )

inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, vector) -> matrix, (vector, matrix) -> matrix, (vector, vector) -> matrix, (row vector, row vector) -> row_vector.

Template Parameters

T_a	type of the first matrix
T_b	type of the second matrix

Parameters

a	First matrix.
b	Second matrix.

Returns

Result of appending the first matrix followed by the second matrix side by side.

Definition at line 42 of file append_col.hpp.

template<typename Scal , typename RowVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_row_vector< RowVec >> * = nullptr>

Eigen::Matrix<return_type_t<Scal, RowVec>, 1, Eigen::Dynamic> stan::math::append_col ( const Scal &  A, const RowVec &  B  )

inline

Return the result of stacking an scalar on top of the a row vector, with the result being a row vector.

This function applies to (scalar, row vector) and returns a row vector.

Template Parameters

Scal	type of the scalar
RowVec	type of the row vector

Parameters

A	scalar.
B	row vector.

Returns

Result of stacking the scalar on top of the row vector.

Definition at line 71 of file append_col.hpp.

template<typename Scal , typename RowVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_row_vector< RowVec >> * = nullptr>

auto stan::math::append_col ( const Scal &  A, const var_value< RowVec > &  B  )

inline

Return the result of stacking an scalar on top of the a row vector, with the result being a row vector.

This function applies to (scalar, row vector) and returns a row vector.

Template Parameters

Scal	type of the scalar
RowVec	A var_value with an inner type of row vector.

Parameters

A	scalar.
B	row vector.

Returns

Result of stacking the scalar on top of the row vector.

Definition at line 81 of file append_col.hpp.

template<typename RowVec , typename Scal , require_t< is_eigen_row_vector< RowVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>

Eigen::Matrix<return_type_t<RowVec, Scal>, 1, Eigen::Dynamic> stan::math::append_col ( const RowVec &  A, const Scal &  B  )

inline

Return the result of stacking a row vector on top of the an scalar, with the result being a row vector.

This function applies to (row vector, scalar) and returns a row vector.

Template Parameters

RowVec	type of the row vector
Scal	type of the scalar

Parameters

A	row vector.
B	scalar.

Returns

Result of stacking the row vector on top of the scalar.

Definition at line 99 of file append_col.hpp.

template<typename RowVec , typename Scal , require_t< is_eigen_row_vector< RowVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>

auto stan::math::append_col ( const var_value< RowVec > &  A, const Scal &  B  )

inline

Return the result of stacking a row vector on top of the an scalar, with the result being a row vector.

This function applies to (row vector, scalar) and returns a row vector.

Template Parameters

RowVec	A var_value with an inner type of row vector.
Scal	type of the scalar

Parameters

A	row vector.
B	scalar.

Returns

Result of stacking the row vector on top of the scalar.

Definition at line 121 of file append_col.hpp.

template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr>

auto stan::math::append_row ( const T1 &  A, const T2 &  B  )

inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, row_vector) -> matrix, (row_vector, matrix) -> matrix, (row_vector, row_vector) -> matrix, (vector, vector) -> vector.

Template Parameters

T1	type of the first matrix
T2	type of the second matrix

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of stacking first matrix on top of second.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, row_vector) -> matrix, (row_vector, matrix) -> matrix, (row_vector, row_vector) -> matrix, (vector, vector) -> vector.

Template Parameters

T1	A var_value with inner matrix type
T1	A var_value with inner matrix type

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of stacking first matrix on top of second.

Definition at line 31 of file append_row.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::append_row ( T_a &&  a, T_b &&  b  )

inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

Given input types result in following outputs: (matrix, matrix) -> matrix, (matrix, row_vector) -> matrix, (row_vector, matrix) -> matrix, (row_vector, row_vector) -> matrix, (vector, vector) -> vector.

Template Parameters

T_a	type of the first matrix
T_b	type of the second matrix

Parameters

a	First matrix.
b	Second matrix.

Returns

Result of stacking first matrix on top of second.

Definition at line 40 of file append_row.hpp.

template<typename Scal , typename ColVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_col_vector< ColVec >> * = nullptr>

Eigen::Matrix<return_type_t<Scal, ColVec>, Eigen::Dynamic, 1> stan::math::append_row ( const Scal &  A, const ColVec &  B  )

inline

Return the result of stacking an scalar on top of the a vector, with the result being a vector.

This function applies to (scalar, vector) and returns a vector.

Template Parameters

Scal	type of the scalar
ColVec	type of the vector

Parameters

A	scalar.
B	vector.

Returns

Result of stacking the scalar on top of the vector.

Definition at line 68 of file append_row.hpp.

template<typename Scal , typename ColVec , require_stan_scalar_t< Scal > * = nullptr, require_t< is_eigen_col_vector< ColVec >> * = nullptr>

auto stan::math::append_row ( const Scal &  A, const var_value< ColVec > &  B  )

inline

Return the result of stacking an scalar on top of the a vector, with the result being a vector.

This function applies to (scalar, vector) and returns a vector.

Template Parameters

Scal	type of the scalar
ColVec	A var_value with inner column vector type.

Parameters

A	scalar.
B	vector.

Returns

Result of stacking the scalar on top of the vector.

Definition at line 78 of file append_row.hpp.

template<typename ColVec , typename Scal , require_t< is_eigen_col_vector< ColVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>

Eigen::Matrix<return_type_t<ColVec, Scal>, Eigen::Dynamic, 1> stan::math::append_row ( const ColVec &  A, const Scal &  B  )

inline

Return the result of stacking a vector on top of the an scalar, with the result being a vector.

This function applies to (vector, scalar) and returns a vector.

Template Parameters

ColVec	type of the vector
Scal	type of the scalar

Parameters

A	vector.
B	scalar.

Returns

Result of stacking the vector on top of the scalar.

Definition at line 95 of file append_row.hpp.

template<typename ColVec , typename Scal , require_t< is_eigen_col_vector< ColVec >> * = nullptr, require_stan_scalar_t< Scal > * = nullptr>

auto stan::math::append_row ( const var_value< ColVec > &  A, const Scal &  B  )

inline

Return the result of stacking a vector on top of the an scalar, with the result being a vector.

This function applies to (vector, scalar) and returns a vector.

Template Parameters

ColVec	a var_value with inner column vector type.
Scal	type of the scalar

Parameters

A	vector.
B	scalar.

Returns

Result of stacking the vector on top of the scalar.

Definition at line 117 of file append_row.hpp.

template<class F , class Tuple >

decltype(auto) constexpr stan::math::apply	(	F &&	f,
		Tuple &&	t
	)

Definition at line 43 of file apply.hpp.

template<typename T1 , typename T2 , typename F , require_all_stan_scalar_t< T1, T2 > * = nullptr>

auto stan::math::apply_scalar_binary ( const T1 &  x, const T2 &  y, const F &  f  )

inline

Base template function for vectorization of binary scalar functions defined by applying a functor to a combination of scalars, containers of matching sizes, or a combination of a scalar and a container.

Specialisation for use with two nested containers (std::vectors).

Specialisation for use with (non-nested) std::vectors.

Specialisation for use with two Eigen inputs.

These containers can be a standard library vector, Eigen dense matrix expression template, or container of these. For each specialisation, the same type as the largest (dimension) input is returned.

This base template function takes (and returns) scalars.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	First input to which operation is applied.
y	Second input to which operation is applied.
f	functor to apply to inputs.

Returns

Scalar with result of applying functor to input.

Eigen's binaryExpr framework is used for more efficient indexing of both row- and column-major inputs without separate loops.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	First Eigen input to which operation is applied.
y	Second Eigen input to which operation is applied.
f	functor to apply to Eigen input.

Returns

Eigen object with result of applying functor to inputs.

Inputs are mapped to Eigen column vectors and then the result is evaluated directly into the returned std::vector (via Eigen::Map).

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of first std::vector to which functor is applied.
T2	Type of second std::vector to which functor is applied.
F	Type of functor to apply.

Parameters

x	First std::vector input to which operation is applied.
y	Second std::vector input to which operation is applied.
f	functor to apply to std::vector inputs.

Returns

std::vector with result of applying functor to inputs.

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of first std::vector to which functor is applied.
T2	Type of second std::vector to which functor is applied.
F	Type of functor to apply.

Parameters

x	First std::vector input to which operation is applied.
y	Second std::vector input to which operation is applied.
f	functor to apply to std::vector inputs.

Returns

std::vector with result of applying functor to inputs.

Definition at line 37 of file apply_scalar_binary.hpp.

template<typename T1 , typename T2 , typename F , require_eigen_vector_vt< is_stan_scalar, T1 > * = nullptr, require_std_vector_vt< std::is_integral, T2 > * = nullptr>

auto stan::math::apply_scalar_binary ( const T1 &  x, const T2 &  y, const F &  f  )

inline

Specialisation for use with one Eigen vector (row or column) and a one-dimensional std::vector of integer types.

Specialisation for use when the one input is an `var_value<Eigen> type and the other is a scalar.

Specialisation for use with one var_value<Eigen vector> (row or column) and a one-dimensional std::vector of integer types.

Specialisation for use with combinations of Eigen::Matrix and var_value<Eigen::Matrix> inputs.

Specialisation for use when the first input is a scalar and the second is a nested std::vector.

Specialisation for use when the first input is a nested std::vector and the second is a scalar.

Specialisation for use when the first input is a scalar and the second is a (non-nested) std::vector.

Specialisation for use when the first input is a (non-nested) std::vector and the second is a scalar.

Specialisation for use when the first input is an scalar and the second is an Eigen type.

Specialisation for use when the first input is an Eigen type and the second is a scalar.

Specialisation for use with a one-dimensional std::vector of integer types and one Eigen vector (row or column).

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Eigen input to which operation is applied.
y	Integer std::vector input to which operation is applied.
f	functor to apply to inputs.

Returns

Eigen object with result of applying functor to inputs.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Integer std::vector input to which operation is applied.
y	Eigen input to which operation is applied.
f	functor to apply to inputs.

Returns

Eigen object with result of applying functor to inputs.

Eigen's unaryExpr framework is used for more efficient indexing of both row- and column-major inputs. The unaryExpr framework also allows for the scalar to be captured and applied to each element in the Eigen object.

Template Parameters

T1	Type of Eigen object to which functor is applied.
T2	Type of scalar to which functor is applied.
F	Type of functor to apply.

Parameters

x	Eigen input to which operation is applied.
y	Scalar input to which operation is applied.
f	functor to apply to Eigen and scalar inputs.

Returns

Eigen object with result of applying functor to inputs.

Note: The return expresssion needs to be evaluated, otherwise the captured function and scalar fall out of scope.

Eigen's unaryExpr framework is used for more efficient indexing of both row- and column-major inputs. The unaryExpr framework also allows for the scalar to be captured and applied to each element in the Eigen object.

Template Parameters

T1	Type of scalar to which functor is applied.
T2	Type of Eigen object to which functor is applied.
F	Type of functor to apply.

Parameters

x	Scalar input to which operation is applied.
y	Eigen input to which operation is applied.
f	functor to apply to Eigen and scalar inputs.

Returns

Eigen object with result of applying functor to inputs.

Note: The return expresssion needs to be evaluated, otherwise the captured function and scalar fall out of scope.

The std::vector input is mapped to an Eigen column vector and then the result is evaluated directly into the returned std::vector (via Eigen::Map).

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of std::vector to which functor is applied.
T2	Type of scalar to which functor is applied.
F	Type of functor to apply.

Parameters

x	std::vector input to which operation is applied.
y	Scalar input to which operation is applied.
f	functor to apply to std::vector and scalar inputs.

Returns

std::vector with result of applying functor to inputs.

The std::vector input is mapped to an Eigen column vector and then the result is evaluated directly into the returned std::vector (via Eigen::Map).

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of scalar to which functor is applied.
T2	Type of std::vector to which functor is applied.
F	Type of functor to apply.

Parameters

x	Scalar input to which operation is applied.
y	std::vector input to which operation is applied.
f	functor to apply to std::vector and scalar inputs.

Returns

std::vector with result of applying functor to inputs.

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of std::vector to which functor is applied.
T2	Type of scalar to which functor is applied.
F	Type of functor to apply.

Parameters

x	std::vector input to which operation is applied.
y	Scalar input to which operation is applied.
f	functor to apply to inputs.

Returns

std::vector with result of applying functor to inputs.

The returned scalar type is deduced to allow for cases where the input and return scalar types differ (e.g., functions implicitly promoting integers).

Template Parameters

T1	Type of scalar to which functor is applied.
T2	Type of std::vector to which functor is applied.
F	Type of functor to apply.

Parameters

x	Scalar input to which operation is applied.
y	std::vector input to which operation is applied.
f	functor to apply to inputs.

Returns

std::vector with result of applying functor to inputs.

Eigen's binaryExpr framework is used for more efficient indexing of both row- and column-major inputs without separate loops.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	First Matrix input to which operation is applied.
y	Second Matrix input to which operation is applied.
f	functor to apply to Matrix inputs.

Returns

var_value<Matrix> with result of applying functor to inputs.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Matrix input to which operation is applied.
y	Integer std::vector input to which operation is applied.
f	functor to apply to inputs.

Returns

var_value<Eigen> object with result of applying functor to inputs.

Template Parameters

T1	Type of either var_value<Matrix> or scalar object to which functor is applied.
T2	Type of either var_value<Matrix> or scalar object to which functor is applied.
F	Type of functor to apply.

Parameters

x	Matrix or Scalar input to which operation is applied.
x	Matrix or Scalar input to which operation is applied.
f	functor to apply to var matrix and scalar inputs.

Returns

`var_value<Matrix> object with result of applying functor to inputs.

Definition at line 76 of file apply_scalar_binary.hpp.

template<typename T1 , typename T2 , typename F , require_eigen_matrix_dynamic_vt< is_stan_scalar, T1 > * = nullptr, require_std_vector_vt< is_std_vector, T2 > * = nullptr, require_std_vector_st< std::is_integral, T2 > * = nullptr>

auto stan::math::apply_scalar_binary ( const T1 &  x, const T2 &  y, const F &  f  )

inline

Specialisation for use with one Eigen matrix and a two-dimensional std::vector of integer types.

Specialisation for use with a two-dimensional std::vector of integer types and one var_value<Matrix>.

Specialisation for use with a two-dimensional std::vector of integer types and one Eigen matrix.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Eigen matrix input to which operation is applied.
y	Nested integer std::vector input to which operation is applied.
f	functor to apply to inputs.

Returns

Eigen object with result of applying functor to inputs.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Nested integer std::vector input to which operation is applied.
y	Eigen matrix input to which operation is applied.
f	functor to apply to inputs.

Returns

Eigen object with result of applying functor to inputs.

Template Parameters

T1	Type of first argument to which functor is applied.
T2	Type of second argument to which functor is applied.
F	Type of functor to apply.

Parameters

x	Either a var matrix or nested integer std::vector input to which operation is applied.
x	Either a var matrix or nested integer std::vector input to which operation is applied.
f	functor to apply to inputs.

Returns

Eigen object with result of applying functor to inputs.

Definition at line 121 of file apply_scalar_binary.hpp.

var stan::math::arg ( const std::complex< var > &  z )

inline

Return the phase angle of the complex argument.

Parameters

[in]

z

argument

Returns

phase angle of the argument

Definition at line 18 of file arg.hpp.

template<typename T >

fvar<T> stan::math::arg ( const std::complex< fvar< T >> &  z )

inline

Return the phase angle of the complex argument.

Template Parameters

T	value type of autodiff variable

Parameters

[in]

z

argument

Returns

phase angle of the argument

Definition at line 19 of file arg.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T stan::math::as_array_or_scalar ( T &&  v )

inline

Returns specified input value.

Template Parameters

T	Type of element.

Parameters

v	Specified value.

Returns

Same value.

Definition at line 19 of file as_array_or_scalar.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::as_array_or_scalar ( T &&  v )

inline

Converts a var_value<T> with inner Eigen matrix type to an var_value<T> with an inner array.

Template Parameters

T	Type of var_value<T> with inner Eigen::Matrix or expression

Parameters

v	Specified var_value<T> with inner Eigen::Matrix or expression.

Returns

var_value<> with Matrix converted to an array.

Definition at line 20 of file as_array_or_scalar.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T& stan::math::as_array_or_scalar ( T &  v )

inline

Returns a reference to rvalue specified input value.

Template Parameters

T	Type of element.

Parameters

v	Specified value.

Returns

Same value.

Definition at line 31 of file as_array_or_scalar.hpp.

template<typename T , typename = require_eigen_t<T>, require_not_eigen_array_t< T > * = nullptr>

auto stan::math::as_array_or_scalar ( T &&  v )

inline

Converts a matrix type to an array.

Converts an std::vector<std::vector> to an Eigen Array.

Converts a std::vector type to an array.

Template Parameters

T	Type of Eigen Matrix or expression

Parameters

v	Specified Eigen Matrix or expression.

Returns

Matrix converted to an array.

Template Parameters

T	Type of scalar element.

Parameters

v	Specified vector.

Returns

Matrix converted to an array.

Template Parameters

T	A standard vector with inner container of a standard vector with an inner stan scalar.

Parameters

v	specified vector of vectorised

Returns

An Eigen Array with dynamic rows and columns.

Definition at line 56 of file as_array_or_scalar.hpp.

int stan::math::as_bool ( const var &  v )

inline

Return 1 if the argument is unequal to zero and 0 otherwise.

Parameters

v

Value.

Returns

1 if argument is equal to zero (or NaN) and 0 otherwise.

Definition at line 16 of file as_bool.hpp.

template<typename T >

bool stan::math::as_bool ( const T &  x )

inline

Return true if the argument is not equal to zero (in the != operator sense) and false otherwise.

Template Parameters

T	type of scalar

Parameters

x

value

Returns

true if value is not equal to zero

Definition at line 17 of file as_bool.hpp.

template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>

auto stan::math::as_column_vector_or_scalar ( const T &  m )

inline

Converts kernel generator expression row or column vector to a column vector.

Template Parameters

T	kernel generator expression.

Parameters

m	Specified input.

Returns

input converted to a column vector.

Definition at line 22 of file as_column_vector_or_scalar.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T stan::math::as_column_vector_or_scalar ( const T &  a )

inline

no-op that passes the scalar

Template Parameters

T	Type of scalar element.

Parameters

a	Specified scalar.

Returns

the scalar.

Definition at line 22 of file as_column_vector_or_scalar.hpp.

template<typename T , typename S >

internal::empty_broadcast_array<T, S, void>& stan::math::as_column_vector_or_scalar

(

internal::empty_broadcast_array< T, S, void > &

a

)

No-op used when working with operands and partials.

This is not implimented so it cannot be invoked and only exists so the compiler can resolve it's output type.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

T&& stan::math::as_column_vector_or_scalar ( T &&  a )

inline

no-op that returns a column vector.

Template Parameters

T	Type inheriting from EigenBase with dynamic compile time rows and fixed column of 1.

Parameters

a	Specified vector.

Returns

Same vector.

Definition at line 44 of file as_column_vector_or_scalar.hpp.

template<typename T , require_eigen_row_vector_t< T > * = nullptr, require_not_eigen_col_vector_t< T > * = nullptr>

auto stan::math::as_column_vector_or_scalar ( T &&  a )

inline

Converts a row vector to an eigen column vector.

Converts var_value with row vector inner type to a var_value<> with inner column vector type.

For row vectors this returns a Transpose<Eigen::Matrix<T, 1, -1>>.

Template Parameters

T	Type inheriting from EigenBase with dynamic compile time columns and fixed row of 1.

Parameters

a	Specified vector.

Returns

Transposed vector.

Template Parameters

T	A var_value<> with an inner row vector type.

Parameters

a	Specified vector.

Returns

Transposed vector.

Definition at line 59 of file as_column_vector_or_scalar.hpp.

template<typename T >

auto stan::math::as_value_array_or_scalar ( T &&  v )

inline

Extract the value from an object.

For eigen types and std::vectors convert to an eigen array and for scalars return a scalar.

Template Parameters

T	A stan scalar, eigen vector, or std::vector.

Parameters

v	Specified value.

Returns

Same value.

Definition at line 21 of file as_value_array_or_scalar.hpp.

template<typename T >

auto stan::math::as_value_column_array_or_scalar ( T &&  a )

inline

Extract the value from an object and for eigen vectors and std::vectors convert to an eigen column array and for scalars return a scalar.

Template Parameters

T	A stan scalar, eigen vector, or std::vector.

Parameters

a	Specified scalar.

Returns

the scalar.

Definition at line 22 of file as_value_column_array_or_scalar.hpp.

template<typename T >

auto stan::math::as_value_column_vector_or_scalar ( T &&  a )

inline

Extract values from input argument and transform to a column vector.

For scalar this returns a scalar. For arithmetic types this is an identity function.

Template Parameters

T	Type of scalar element.

Parameters

a	Specified scalar.

Returns

the scalar.

Definition at line 23 of file as_value_column_vector_or_scalar.hpp.

template<typename T >

fvar<T> stan::math::asin ( const fvar< T > &  x )

inline

Definition at line 15 of file asin.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::asin ( const var_value< T > &  A )

inline

Returns the elementwise asin() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise asin() of the input, in radians.

Definition at line 21 of file asin.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::asin ( const std::complex< fvar< T >> &  z )

inline

Return the arc sine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

arc sine of the argument

Definition at line 29 of file asin.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::asin ( const Container &  x )

inline

Returns the elementwise asin() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

Arcsine of each variable in the container, in radians.

Definition at line 47 of file asin.hpp.

var stan::math::asin ( const var &  x )

inline

Return the principal value of the arc sine, in radians, of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}\).

\[ \mbox{asin}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arcsin(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asin}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arcsin(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arcsin(x)}{\partial x} = \frac{1}{\sqrt{1-x^2}} \]

Parameters

x	Variable in range [-1, 1].

Returns

Arc sine of variable, in radians.

Definition at line 53 of file asin.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::asin ( const Container &  x )

inline

Version of asin() that accepts std::vectors, Eigen Matrix/Array objects, or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Arcsine of each variable in the container, in radians.

Definition at line 61 of file asin.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::asin ( const VarMat &  x )

inline

Return the principal value of the arc sine, in radians, of the specified variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

x	Variable with cells in range [-1, 1].

Returns

Arc sine of variable, in radians.

Definition at line 68 of file asin.hpp.

std::complex<var> stan::math::asin ( const std::complex< var > &  z )

inline

Return the arc sine of the complex argument.

Parameters

[in]

z

argument

Returns

arc sine of the argument

Definition at line 82 of file asin.hpp.

template<typename T >

fvar<T> stan::math::asinh ( const fvar< T > &  x )

inline

Definition at line 15 of file asinh.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::asinh ( const var_value< T > &  A )

inline

Returns the elementwise asinh() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise asinh() of the input, in radians.

Definition at line 21 of file asinh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::asinh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic arcsine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic arcsine of the argument

Definition at line 28 of file asinh.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::asinh ( const T &  x )

inline

Returns the elementwise asinh() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

T	type of container

Parameters

x

container

Returns

Inverse hyperbolic sine of each value in the container.

Definition at line 49 of file asinh.hpp.

var stan::math::asinh ( const var &  x )

inline

The inverse hyperbolic sine function for variables (C99).

The derivative is defined by

\(\frac{d}{dx} \mbox{asinh}(x) = \frac{x}{x^2 + 1}\).

\[ \mbox{asinh}(x) = \begin{cases} \sinh^{-1}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} \frac{\partial\, \sinh^{-1}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right) \]

\[ \frac{\partial \, \sinh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2+1}} \]

Parameters

x	The variable.

Returns

Inverse hyperbolic sine of the variable.

Definition at line 59 of file asinh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::asinh ( const VarMat &  x )

inline

The inverse hyperbolic sine function for variables (C99).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

x	The variable.

Returns

Inverse hyperbolic sine of the variable.

Definition at line 73 of file asinh.hpp.

std::complex<var> stan::math::asinh ( const std::complex< var > &  z )

inline

Return the hyperbolic arcsine of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic arcsine of the argument

Definition at line 88 of file asinh.hpp.

template<typename T_lhs , typename T_rhs , require_all_stan_scalar_t< T_lhs, T_rhs > * = nullptr>

void stan::math::assign ( T_lhs &  x, const T_rhs &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will match arguments where the right-hand side is assignable to the left and they are not both std::vector or Eigen::Matrix types.

Template Parameters

T_lhs	Type of left-hand side.
T_rhs	Type of right-hand side.

Parameters

x	Left-hand side.
y	Right-hand side.

Definition at line 46 of file assign.hpp.

template<typename T_lhs , typename T_rhs , require_all_eigen_t< T_lhs, T_rhs > * = nullptr>

void stan::math::assign ( T_lhs &&  x, const T_rhs &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types.

Template Parameters

T_lhs	type of the left-hand side matrix
T_rhs	type of the right-hand side matrix

Parameters

x	Left-hand side matrix.
y	Right-hand side matrix.

Exceptions

std::invalid_argument

if sizes do not match.

Definition at line 67 of file assign.hpp.

template<typename T_lhs , typename T_rhs >

void stan::math::assign ( std::vector< T_lhs > &  x, const std::vector< T_rhs > &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both std::vector, and will call assign() element-by element.

For example, a std::vector<int> can be assigned to a std::vector<double> using this function.

Template Parameters

T_lhs	type of elements in the left-hand side vector
T_rhs	type of elements in the right-hand side vector

Parameters

x	Left-hand side vector.
y	Right-hand side vector.

Exceptions

std::invalid_argument

if sizes do not match.

Definition at line 92 of file assign.hpp.

template<typename T >

fvar<T> stan::math::atan ( const fvar< T > &  x )

inline

Definition at line 15 of file atan.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::atan ( const var_value< T > &  A )

inline

Returns the elementwise atan() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise atan() of the input, in radians.

Definition at line 21 of file atan.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::atan ( const std::complex< fvar< T >> &  z )

inline

Return the arc tangent of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

arc tanget of the argument

Definition at line 28 of file atan.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::atan ( const Container &  x )

inline

Returns the elementwise atan() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

Arctan of each value in x, in radians.

Definition at line 45 of file atan.hpp.

var stan::math::atan ( const var &  x )

inline

Return the principal value of the arc tangent, in radians, of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}\).

\[ \mbox{atan}(x) = \begin{cases} \arctan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan}(x)}{\partial x} = \begin{cases} \frac{\partial\, \arctan(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arctan(x)}{\partial x} = \frac{1}{x^2+1} \]

Parameters

x	Variable in range [-1, 1].

Returns

Arc tangent of variable, in radians.

Definition at line 54 of file atan.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::atan ( const Container &  x )

inline

Version of atan() that accepts std::vectors, Eigen Matrix/Array objects, or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Elementwise atan of members of container.

Definition at line 59 of file atan.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::atan ( const VarMat &  x )

inline

Return the principal value of the arc tangent, in radians, of the specified variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

x	Variable in range [-1, 1].

Returns

Arc tangent of variable, in radians.

Definition at line 70 of file atan.hpp.

std::complex<var> stan::math::atan ( const std::complex< var > &  z )

inline

Return the arc tangent of the complex argument.

Parameters

[in]

z

argument

Returns

arc tangent of the argument

Definition at line 83 of file atan.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 13 of file atan2.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 21 of file atan2.hpp.

var stan::math::atan2 ( const var &  a, const var &  b  )

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath).

The partial derivatives are defined by

\( \frac{\partial}{\partial x} \arctan \frac{x}{y} = \frac{y}{x^2 + y^2}\), and

\( \frac{\partial}{\partial y} \arctan \frac{x}{y} = \frac{-x}{x^2 + y^2}\).

Parameters

a	Numerator variable.
b	Denominator variable.

Returns

The arc tangent of the fraction, in radians.

Definition at line 27 of file atan2.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 28 of file atan2.hpp.

var stan::math::atan2 ( const var &  a, double  b  )

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath).

The derivative with respect to the variable is

\( \frac{d}{d x} \arctan \frac{x}{c} = \frac{c}{x^2 + c^2}\).

Parameters

a	Numerator variable.
b	Denominator scalar.

Returns

The arc tangent of the fraction, in radians.

Definition at line 48 of file atan2.hpp.

var stan::math::atan2 ( double  a, const var &  b  )

inline

Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath).

The derivative with respect to the variable is

\( \frac{\partial}{\partial y} \arctan \frac{c}{y} = \frac{-c}{c^2 + y^2}\).

\[ \mbox{atan2}(x, y) = \begin{cases} \arctan\left(\frac{x}{y}\right) & \mbox{if } -\infty\leq x \leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial x} = \begin{cases} \frac{y}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial y} = \begin{cases} -\frac{x}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	Numerator scalar.
b	Denominator variable.

Returns

The arc tangent of the fraction, in radians.

Definition at line 92 of file atan2.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::atanh ( const var_value< T > &  A )

inline

Returns the elementwise atanh() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise atanh() of the input, in radians.

Definition at line 21 of file atanh.hpp.

template<typename T >

fvar<T> stan::math::atanh ( const fvar< T > &  x )

inline

Return inverse hyperbolic tangent of specified value.

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

Argument.

Returns

Inverse hyperbolic tangent of argument.

Exceptions

std::domain_error

if x < -1 or x > 1.

Definition at line 24 of file atanh.hpp.

double stan::math::atanh ( double  x )

inline

Return the inverse hyperbolic tangent of the specified value.

An argument of -1 returns negative infinity and an argument of 1 returns infinity. Returns nan for nan argument.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic tangent of the argument.

Exceptions

std::domain_error

If argument is not in [-1, 1].

Definition at line 27 of file atanh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::atanh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic arc tangent of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic arc tangent of the argument

Definition at line 36 of file atanh.hpp.

double stan::math::atanh ( int  x )

inline

Integer version of atanh.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic tangent of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 43 of file atanh.hpp.

var stan::math::atanh ( const var &  x )

inline

The inverse hyperbolic tangent function for variables (C99).

The derivative is defined by

\(\frac{d}{dx} \mbox{atanh}(x) = \frac{1}{1 - x^2}\).

\[ \mbox{atanh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \tanh^{-1}(x) & \mbox{if } -1\leq x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atanh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \tanh^{-1}(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \tanh^{-1}(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \]

\[ \frac{\partial \, \tanh^{-1}(x)}{\partial x} = \frac{1}{1-x^2} \]

Parameters

x	The variable.

Returns

Inverse hyperbolic tangent of the variable.

Exceptions

std::domain_error

if a < -1 or a > 1

Definition at line 58 of file atanh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::atanh ( const VarMat &  x )

inline

The inverse hyperbolic tangent function for variables (C99).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

x	The variable.

Returns

Inverse hyperbolic tangent of the variable.

Exceptions

std::domain_error

if a < -1 or a > 1

Definition at line 73 of file atanh.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::atanh ( const T &  x )

inline

Return the elementwise application of atanh() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise atanh of members of container.

Definition at line 78 of file atanh.hpp.

std::complex<var> stan::math::atanh ( const std::complex< var > &  z )

inline

Return the hyperbolic arc tangent of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic arc tangent of the argument

Definition at line 87 of file atanh.hpp.

template<typename T >

void stan::math::autocorrelation	(	const std::vector< T > &	y,
		std::vector< T > &	ac,
		Eigen::FFT< T > &	fft
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.
fft	FFT engine instance.

Definition at line 62 of file autocorrelation.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocorrelation	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	ac,
		Eigen::FFT< T > &	fft
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.
fft	FFT engine instance.

Definition at line 119 of file autocorrelation.hpp.

template<typename T >

void stan::math::autocorrelation	(	const std::vector< T > &	y,
		std::vector< T > &	ac
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocorrelation function.

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.

Definition at line 164 of file autocorrelation.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocorrelation	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	ac
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocorrelation function

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.

Definition at line 191 of file autocorrelation.hpp.

template<typename T >

void stan::math::autocovariance	(	const std::vector< T > &	y,
		std::vector< T > &	acov,
		Eigen::FFT< T > &	fft
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariance.
fft	FFT engine instance.

Definition at line 33 of file autocovariance.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocovariance	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	acov,
		Eigen::FFT< T > &	fft
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	scalar type
DerivedA	type of the first matrix
DerivedB	type of the second matrix

Parameters

y	Input sequence.
acov	Autocovariance.
fft	FFT engine instance.

Definition at line 66 of file autocovariance.hpp.

template<typename T >

void stan::math::autocovariance	(	const std::vector< T > &	y,
		std::vector< T > &	acov
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocovariance function.

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariances.

Definition at line 89 of file autocovariance.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocovariance	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	acov
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocovariance function

Template Parameters

T	scalar type
DerivedA	type of the first matrix
DerivedB	type of the second matrix

Parameters

y	Input sequence.
acov	Autocovariances.

Definition at line 118 of file autocovariance.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta >

return_type_t<T_x, T_beta, T_alpha> stan::math::bernoulli_logit_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 166 of file bernoulli_logit_glm_lpmf.hpp.

template<typename T_n , typename T_prob >

return_type_t<T_prob> stan::math::bernoulli_logit_lpmf ( const T_n &  n, const T_prob &  theta  )

inline

Definition at line 92 of file bernoulli_logit_lpmf.hpp.

template<typename T_y , typename T_prob >

return_type_t<T_prob> stan::math::bernoulli_lpmf ( const T_y &  n, const T_prob &  theta  )

inline

Definition at line 115 of file bernoulli_lpmf.hpp.

var stan::math::bessel_first_kind ( int  v, const var &  a  )

inline

Definition at line 11 of file bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::bessel_first_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file bessel_first_kind.hpp.

template<typename T1 , typename T2 , require_st_integral< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr>

auto stan::math::bessel_first_kind ( const T1 &  v, const var_value< T2 > &  a  )

inline

Overload with var_value<Matrix> for int, std::vector<int>, and std::vector<std::vector<int>>

Definition at line 27 of file bessel_first_kind.hpp.

template<typename T2 , require_arithmetic_t< T2 > * = nullptr>

T2 stan::math::bessel_first_kind ( int  v, const T2  z  )

inline

\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_first\_kind}(v, x)}{\partial x} = \begin{cases} \frac{\partial\, J_v(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ J_v(x)=\left(\frac{1}{2}x\right)^v \sum_{k=0}^\infty \frac{\left(-\frac{1}{4}x^2\right)^k}{k!\, \Gamma(v+k+1)} \]

\[ \frac{\partial \, J_v(x)}{\partial x} = \frac{v}{x}J_v(x)-J_{v+1}(x) \]

Definition at line 39 of file bessel_first_kind.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_not_var_matrix_t< T2 > * = nullptr>

auto stan::math::bessel_first_kind ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the bessel first kind function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Bessel first kind function applied to the two inputs.

Definition at line 56 of file bessel_first_kind.hpp.

var stan::math::bessel_second_kind ( int  v, const var &  a  )

inline

Definition at line 11 of file bessel_second_kind.hpp.

template<typename T >

fvar<T> stan::math::bessel_second_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file bessel_second_kind.hpp.

template<typename T1 , typename T2 , require_st_integral< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr>

auto stan::math::bessel_second_kind ( const T1 &  v, const var_value< T2 > &  a  )

inline

Overload with var_value<Matrix> for int, std::vector<int>, and std::vector<std::vector<int>>

Definition at line 26 of file bessel_second_kind.hpp.

template<typename T2 , require_arithmetic_t< T2 > * = nullptr>

T2 stan::math::bessel_second_kind ( int  v, const T2  z  )

inline

\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_second\_kind}(v, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ \frac{\partial\, Y_v(x)}{\partial x} & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)} \]

\[ \frac{\partial \, Y_v(x)}{\partial x} = \frac{v}{x}Y_v(x)-Y_{v+1}(x) \]

Definition at line 41 of file bessel_second_kind.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::bessel_second_kind ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the bessel second kind function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Bessel second kind function applied to the two inputs.

Definition at line 56 of file bessel_second_kind.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::beta ( T_a &&  a, T_b &&  b  )

inline

Return the elementwise beta() on two input kernel generator expression.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

elementwise beta()

Definition at line 31 of file beta.hpp.

var stan::math::beta ( const var &  a, const var &  b  )

inline

Returns the beta function and gradients for two var inputs.

\[ \mathrm{beta}(a,b) = \left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial a} = \left(\psi^{\left(0\right)}\left(a\right) - \psi^{\left(0\right)} \left(a + b\right)\right) \mathrm{beta}(a,b) \]

\[ \frac{\partial }{\partial b} = \left(\psi^{\left(0\right)}\left(b\right) - \psi^{\left(0\right)} \left(a + b\right)\right) \mathrm{beta}(a,b) \]

Parameters

a	var Argument
b	var Argument

Returns

Result of beta function

Definition at line 37 of file beta.hpp.

template<typename T >

fvar<T> stan::math::beta ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return fvar with the beta function applied to the specified arguments and its gradient.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

\[ \mbox{beta}(\alpha, \beta) = \begin{cases} \int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \left(\psi(\alpha)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \left(\psi(\beta)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T	inner type of the fvar

Parameters

x1	First value
x2	Second value

Returns

Fvar with result beta function of arguments and gradients.

Definition at line 51 of file beta.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::beta ( const T1  a, const T2  b  )

inline

Return the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

\[ \mbox{beta}(\alpha, \beta) = \begin{cases} \int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \left(\psi(\alpha)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \left(\psi(\beta)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first value
T2	type of second value

Parameters

a	First value
b	Second value

Returns

Beta function applied to the two values.

Definition at line 53 of file beta.hpp.

template<typename T >

fvar<T> stan::math::beta ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 60 of file beta.hpp.

template<typename T >

fvar<T> stan::math::beta ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 67 of file beta.hpp.

var stan::math::beta ( const var &  a, double  b  )

inline

Returns the beta function and gradient for first var input.

\[ \mathrm{beta}(a,b) = \left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial a} = \left(\psi^{\left(0\right)}\left(a\right) - \psi^{\left(0\right)} \left(a + b\right)\right) \mathrm{beta}(a,b) \]

Parameters

a	var Argument
b	double Argument

Returns

Result of beta function

Definition at line 67 of file beta.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::beta ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the beta function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Beta function applied to the two inputs.

Definition at line 70 of file beta.hpp.

var stan::math::beta ( double  a, const var &  b  )

inline

Returns the beta function and gradient for second var input.

\[ \mathrm{beta}(a,b) = \left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial b} = \left(\psi^{\left(0\right)}\left(b\right) - \psi^{\left(0\right)} \left(a + b\right)\right) \mathrm{beta}(a,b) \]

Parameters

a	double Argument
b	var Argument

Returns

Result of beta function

Definition at line 93 of file beta.hpp.

template<typename Mat1 , typename Mat2 , require_any_var_matrix_t< Mat1, Mat2 > * = nullptr, require_all_matrix_t< Mat1, Mat2 > * = nullptr>

auto stan::math::beta ( const Mat1 &  a, const Mat2 &  b  )

inline

Definition at line 104 of file beta.hpp.

template<typename Scalar , typename VarMat , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr>

auto stan::math::beta ( const Scalar &  a, const VarMat &  b  )

inline

Definition at line 150 of file beta.hpp.

template<typename VarMat , typename Scalar , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr>

auto stan::math::beta ( const VarMat &  a, const Scalar &  b  )

inline

Definition at line 193 of file beta.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >

return_type_t<T_size1, T_size2> stan::math::beta_binomial_lpmf	(	const T_n &	n,
		const T_N &	N,
		const T_size1 &	alpha,
		const T_size2 &	beta
	)

Definition at line 171 of file beta_binomial_lpmf.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >

return_type_t<T_y, T_scale_succ, T_scale_fail> stan::math::beta_lpdf ( const T_y &  y, const T_scale_succ &  alpha, const T_scale_fail &  beta  )

inline

Definition at line 126 of file beta_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_prec >

return_type_t<T_y, T_loc, T_prec> stan::math::beta_proportion_lpdf ( const T_y &  y, const T_loc &  mu, const T_prec &  kappa  )

inline

Definition at line 128 of file beta_proportion_lpdf.hpp.

template<typename T >

fvar<T> stan::math::binary_log_loss ( int  y, const fvar< T > &  y_hat  )

inline

Definition at line 12 of file binary_log_loss.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

T stan::math::binary_log_loss ( int  y, const T &  y_hat  )

inline

Returns the log loss function for binary classification with specified reference and response values.

The log loss function for prediction \(\hat{y} \in [0, 1]\) given outcome \(y \in \{ 0, 1 \}\) is

\(\mbox{logloss}(1, \hat{y}) = -\log \hat{y} \), and

\(\mbox{logloss}(0, \hat{y}) = -\log (1 - \hat{y}) \).

Template Parameters

T	value type

Parameters

[in]	y	reference value, either 0 or 1
[in]	y_hat	response value in [0, 1]

Returns

Log loss for response given reference value

Definition at line 30 of file binary_log_loss.hpp.

var stan::math::binary_log_loss ( int  y, const var &  y_hat  )

inline

The log loss function for variables (stan).

See binary_log_loss() for the double-based version.

The derivative with respect to the variable \(\hat{y}\) is

\(\frac{d}{d\hat{y}} \mbox{logloss}(1, \hat{y}) = - \frac{1}{\hat{y}}\), and

\(\frac{d}{d\hat{y}} \mbox{logloss}(0, \hat{y}) = \frac{1}{1 - \hat{y}}\).

\[ \mbox{binary\_log\_loss}(y, \hat{y}) = \begin{cases} y \log \hat{y} + (1 - y) \log (1 - \hat{y}) & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binary\_log\_loss}(y, \hat{y})}{\partial \hat{y}} = \begin{cases} \frac{y}{\hat{y}}-\frac{1-y}{1-\hat{y}} & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

Parameters

y	Reference value.
y_hat	Response variable.

Returns

Log loss of response versus reference value.

Definition at line 46 of file binary_log_loss.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_not_var_matrix_t< T2 > * = nullptr>

auto stan::math::binary_log_loss ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the binary log loss function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Binary log loss function applied to the two inputs.

Definition at line 47 of file binary_log_loss.hpp.

template<typename Mat , require_eigen_t< Mat > * = nullptr>

auto stan::math::binary_log_loss ( int  y, const var_value< Mat > &  y_hat  )

inline

Overload with int and var_value<Matrix>

Definition at line 62 of file binary_log_loss.hpp.

template<typename StdVec , typename Mat , require_eigen_t< Mat > * = nullptr, require_st_integral< StdVec > * = nullptr>

auto stan::math::binary_log_loss ( const StdVec &  y, const var_value< Mat > &  y_hat  )

inline

Overload with var_value<Matrix> for std::vector<int> and std::vector<std::vector<int>>

Definition at line 82 of file binary_log_loss.hpp.

template<typename T_n , typename T_k , require_all_stan_scalar_t< T_n, T_k > * = nullptr>

return_type_t<T_n, T_k> stan::math::binomial_coefficient_log ( const T_n  n, const T_k  k  )

inline

Return the log of the binomial coefficient for the specified arguments.

The binomial coefficient, \({n \choose k}\), read "n choose k", is defined for \(0 \leq k \leq n\) by

\({n \choose k} = \frac{n!}{k! (n-k)!}\).

This function uses Gamma functions to define the log and generalize the arguments to continuous n and k.

\( \log {n \choose k} = \log \ \Gamma(n+1) - \log \Gamma(k+1) - \log \Gamma(n-k+1)\).

\[ \mbox{binomial\_coefficient\_log}(x, y) = \begin{cases} \textrm{error} & \mbox{if } y > x + 1 \textrm{ or } y < -1 \textrm{ or } x < -1\\ \ln\Gamma(x+1) & \mbox{if } -1 < y < x + 1 \\ \quad -\ln\Gamma(y+1)& \\ \quad -\ln\Gamma(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } y > x + 1 \textrm{ or } y < -1 \textrm{ or } x < -1\\ \Psi(x+1) & \mbox{if } 0\leq y \leq x \\ \quad -\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial y} = \begin{cases} \textrm{error} & \mbox{if } y > x + 1 \textrm{ or } y < -1 \textrm{ or } x < -1\\ -\Psi(y+1) & \mbox{if } 0\leq y \leq x \\ \quad +\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

This function is numerically more stable than naive evaluation via lgamma.

Template Parameters

T_n	type of the first argument
T_k	type of the second argument

Parameters

n	total number of objects.
k	number of objects chosen.

Returns

log (n choose k).

Definition at line 81 of file binomial_coefficient_log.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::binomial_coefficient_log ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the binomial coefficient log function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Binomial coefficient log function applied to the two inputs.

Definition at line 164 of file binomial_coefficient_log.hpp.

template<typename T_n , typename T_N , typename T_prob >

return_type_t<T_prob> stan::math::binomial_logit_lpmf ( const T_n &  n, const T_N &  N, const T_prob &  alpha  )

inline

Definition at line 101 of file binomial_logit_lpmf.hpp.

template<typename T_n , typename T_N , typename T_prob >

return_type_t<T_prob> stan::math::binomial_lpmf ( const T_n &  n, const T_N &  N, const T_prob &  theta  )

inline

Definition at line 140 of file binomial_lpmf.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::block ( const T &  m, size_t  i, size_t  j, size_t  nrows, size_t  ncols  )

inline

Return a nrows x ncols submatrix starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
i	Starting row.
j	Starting column.
nrows	Number of rows in block.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 22 of file block.hpp.

template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>

auto stan::math::block ( T_x &&  x, size_t  i, size_t  j, size_t  nrows, size_t  ncols  )

inline

Return a nrows x ncols submatrix starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

x	Matrix.
i	Starting row.
j	Starting column.
nrows	Number of rows in block.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 24 of file block.hpp.

template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>

auto stan::math::block ( const T &  m, size_t  i, size_t  j, size_t  nrows, size_t  ncols  )

inline

Return a nrows x ncols submatrix starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
i	Starting row.
j	Starting column.
nrows	Number of rows in block.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 27 of file block.hpp.

template<int R, int C>

vari** stan::math::build_vari_array

(

const Eigen::Matrix< var, R, C > &

x

)

Allocates and populates a flat array of vari pointers in the autodiff arena with the varis pointed to by the vars in the input Eigen matrix.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

x	input matrix

Returns

Flat array of vari pointers

Definition at line 21 of file build_vari_array.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta >

return_type_t<T_x, T_alpha, T_beta> stan::math::categorical_logit_glm_lpmf	(	const T_y &	y,
		const T_x &	x,
		const T_alpha &	alpha,
		const T_beta &	beta
	)

Definition at line 196 of file categorical_logit_glm_lpmf.hpp.

template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_logit_lpmf	(	int	n,
		const T_prob &	beta
	)

Definition at line 18 of file categorical_logit_lpmf.hpp.

template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_logit_lpmf	(	const std::vector< int > &	ns,
		const T_prob &	beta
	)

Definition at line 36 of file categorical_logit_lpmf.hpp.

template<typename T_n , typename T_prob , require_st_integral< T_n > * = nullptr, require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_logit_lpmf ( const T_n &  ns, const T_prob &  beta  )

inline

Definition at line 66 of file categorical_logit_lpmf.hpp.

template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_lpmf	(	int	n,
		const T_prob &	theta
	)

Definition at line 17 of file categorical_lpmf.hpp.

template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_lpmf	(	const std::vector< int > &	ns,
		const T_prob &	theta
	)

Definition at line 33 of file categorical_lpmf.hpp.

template<typename T_n , typename T_prob , require_st_integral< T_n > * = nullptr, require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::categorical_lpmf ( const T_n &  ns, const T_prob &  theta  )

inline

Definition at line 62 of file categorical_lpmf.hpp.

template<class RNG >

int stan::math::categorical_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  theta, RNG &  rng  )

inline

Definition at line 14 of file categorical_rng.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::cauchy_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 120 of file cauchy_lpdf.hpp.

template<typename T >

fvar<T> stan::math::cbrt ( const fvar< T > &  x )

inline

Return cube root of specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

Argument.

Returns

Cube root of argument.

Definition at line 20 of file cbrt.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::cbrt ( const var_value< T > &  A )

inline

Returns the elementwise cbrt() of the input var_value<matrix_cl<double>>.

Parameters

A	input var_value<matrix_cl<double>>.

Returns

Elementwise cbrt() of the input argument.

Definition at line 21 of file cbrt.hpp.

var stan::math::cbrt ( const var &  a )

inline

Returns the cube root of the specified variable (C99).

The derivative is

\(\frac{d}{dx} x^{1/3} = \frac{1}{3 x^{2/3}}\).

\[ \mbox{cbrt}(x) = \begin{cases} \sqrt[3]{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cbrt}(x)}{\partial x} = \begin{cases} \frac{1}{3x^{2/3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Cube root of the variable.

Definition at line 37 of file cbrt.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::cbrt ( const T &  x )

inline

Returns the elementwise cbrt() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

T	type of container

Parameters

x

container

Returns

Cube root of each value in x.

Definition at line 37 of file cbrt.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::cbrt ( const VarMat &  a )

inline

Returns the cube root of the specified variable (C99).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Specified variable.

Returns

Cube root of the variable.

Definition at line 50 of file cbrt.hpp.

template<typename T >

fvar<T> stan::math::ceil ( const fvar< T > &  x )

inline

Definition at line 12 of file ceil.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::ceil ( const var_value< T > &  A )

inline

Returns the elementwise ceil() of the specified variable.

Parameters

A	input var_value<matrix_cl<double>>.

Returns

Elementwise ceil() of the input argument.

Definition at line 20 of file ceil.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::ceil ( const Container &  x )

inline

Returns the elementwise ceil() of the input, which may be a scalar or any Stan container of numeric scalars.

Version of ceil() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

type of container

Parameters

x

container

Returns

Least integer >= each value in x.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Least integer >= each value in x.

Definition at line 40 of file ceil.hpp.

var stan::math::ceil ( const var &  a )

inline

Return the ceiling of the specified variable (cmath).

The derivative of the ceiling function is defined and zero everywhere but at integers, and we set them to zero for convenience,

\(\frac{d}{dx} {\lceil x \rceil} = 0\).

The ceiling function rounds up. For double values, this is the smallest integral value that is not less than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{ceil}(x) = \begin{cases} \lceil x\rceil & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{ceil}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Ceiling of the variable.

Definition at line 47 of file ceil.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::ceil ( const var_value< T > &  a )

inline

Definition at line 50 of file ceil.hpp.

template<typename T_a1 , typename T_a2 , typename T_b1 , typename T_z >

void stan::math::check_2F1_converges ( const char *  function, const T_a1 &  a1, const T_a2 &  a2, const T_b1 &  b1, const T_z &  z  )

inline

Check if the hypergeometric function (2F1) called with supplied arguments will converge, assuming arguments are finite values.

Template Parameters

T_a1	Type of a1
T_a2	Type of a2
T_b1	Type of b1
T_z	Type of z

Parameters

function	Name of function ultimately relying on 2F1 (for error messages)
a1	Variable to check
a2	Variable to check
b1	Variable to check
z	Variable to check

Exceptions

<code>domain_error</code>

if 2F1(a1, a2, b1, z) does not meet convergence conditions, or if any coefficient is NaN.

Definition at line 35 of file check_2F1_converges.hpp.

template<typename T_a1 , typename T_a2 , typename T_a3 , typename T_b1 , typename T_b2 , typename T_z >

void stan::math::check_3F2_converges ( const char *  function, const T_a1 &  a1, const T_a2 &  a2, const T_a3 &  a3, const T_b1 &  b1, const T_b2 &  b2, const T_z &  z  )

inline

Check if the hypergeometric function (3F2) called with supplied arguments will converge, assuming arguments are finite values.

Template Parameters

T_a1	Type of a1
T_a2	Type of a2
T_a3	Type of a3
T_b1	Type of b1
T_b2	Type of b2
T_z	Type of z

Parameters

function	Name of function ultimately relying on 3F2 (for error & messages)
a1	Variable to check
a2	Variable to check
a3	Variable to check
b1	Variable to check
b2	Variable to check
z	Variable to check

Exceptions

<code>domain_error</code>

if 3F2(a1, a2, a3, b1, b2, z) does not meet convergence conditions, or if any coefficient is NaN.

Definition at line 40 of file check_3F2_converges.hpp.

template<typename T_y , typename T_low , typename T_high >

void stan::math::check_bounded ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, const T_high &  high  )

inline

Check if the value is between the low and high values, inclusively.

Template Parameters

T_y	Type of value
T_low	Type of low value
T_high	Type of high value

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Value to check
low	Low bound
high	High bound

Exceptions

<code>std::domain_error</code>

otherwise. This also throws if any of the arguments are NaN.

Definition at line 75 of file check_bounded.hpp.

template<typename Mat , require_matrix_t< Mat > * = nullptr>

void stan::math::check_cholesky_factor ( const char *  function, const char *  name, const Mat &  y  )

inline

Throw an exception if the specified matrix is not a valid Cholesky factor.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Template Parameters

Mat	Type inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<tt>std::domain_error</tt>

if y is not a valid Cholesky factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 33 of file check_cholesky_factor.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_cholesky_factor	(	const char *	function,
		const char *	name,
		const StdVec &	y
	)

Throw an exception if the specified matrix is not a valid Cholesky factor.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Template Parameters

StdVec

A standard vector with inner type either inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Standard vector of matrices to test

Exceptions

<tt>std::domain_error</tt>

if y is not a valid Cholesky factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 61 of file check_cholesky_factor.hpp.

template<typename Mat , require_matrix_t< Mat > * = nullptr>

void stan::math::check_cholesky_factor_corr	(	const char *	function,
		const char *	name,
		const Mat &	y
	)

Throw an exception if the specified matrix is not a valid Cholesky factor of a correlation matrix.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive and each row has unit Euclidean length. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N). Tolerance is specified by math::CONSTRAINT_TOLERANCE. Tolerance is specified by math::CONSTRAINT_TOLERANCE.

Template Parameters

Mat	Type inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<tt>std::domain_error</tt>

if y is not a valid Cholesky factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 37 of file check_cholesky_factor_corr.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_cholesky_factor_corr	(	const char *	function,
		const char *	name,
		const StdVec &	y
	)

Throw an exception if the specified matrix is not a valid Cholesky factor of a correlation matrix.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive and each row has unit Euclidean length. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N). Tolerance is specified by math::CONSTRAINT_TOLERANCE. Tolerance is specified by math::CONSTRAINT_TOLERANCE.

Template Parameters

StdVec

A standard vector with inner type either inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Standard vector of matrics to test

Exceptions

<tt>std::domain_error</tt>

if y[i] is not a valid Cholesky factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 69 of file check_cholesky_factor_corr.hpp.

template<typename T_y , require_any_t< is_matrix< T_y >, is_prim_or_rev_kernel_expression< T_y >> * = nullptr>

void stan::math::check_column_index ( const char *  function, const char *  name, const T_y &  y, size_t  i  )

inline

Check if the specified index is a valid column of the matrix.

By default this is a 1-indexed check (as opposed to 0-indexed). Behavior can be changed by setting stan::error_index::value. This function will throw an std::out_of_range exception if the index is out of bounds.

Template Parameters

T_y	Type of matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	matrix to test
i	column index to check

Exceptions

<code>std::out_of_range</code>

if index is an invalid column

Definition at line 30 of file check_column_index.hpp.

template<typename T >

void stan::math::check_consistent_size ( const char *  function, const char *  name, const T &  x, size_t  expected_size  )

inline

Check if x is consistent with size expected_size.

x is consistent with size expected_size if x is a vector of size expected_size, or a non vector.

Template Parameters

T	type of value

Parameters

function	function name (for error messages)
name	variable name (for error messages)
x	variable to check for consistent size
expected_size	expected size if x is a vector

Exceptions

<tt>invalid_argument</tt>

if the size is inconsistent

Definition at line 25 of file check_consistent_size.hpp.

void stan::math::check_consistent_sizes ( const char *  )

inline

Trivial no input case, this function is a no-op.

Definition at line 15 of file check_consistent_sizes.hpp.

template<typename T1 >

void stan::math::check_consistent_sizes ( const char *  , const char *  , const T1 &    )

inline

Base case of recursion, this function is a no-op.

Template Parameters

T1	type of first input

Definition at line 22 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename... Ts>

void stan::math::check_consistent_sizes ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2, const Ts &...  names_and_xs  )

inline

Check that the inputs are of consistent size.

Two inputs are of consistent size if they are vectors of the same size, or if at least one is not a vector.

E.g.: check_consistent_sizes("some_function", "x1", x1, "x2", x2, etc.).

Template Parameters

T1	type of first input
T2	type of second input
Ts	type of other inputs

Parameters

function	function name (for error messages)
name1	name of variable corresponding to first input
x1	first input
name2	name of variable corresponding to second input
x2	second input
names_and_xs	more inputs

Exceptions

<tt>invalid_argument</tt>

if sizes are inconsistent

Definition at line 46 of file check_consistent_sizes.hpp.

void stan::math::check_consistent_sizes_mvt ( const char *  )

inline

Trivial no input case, this function is a no-op.

Definition at line 15 of file check_consistent_sizes_mvt.hpp.

template<typename T1 >

void stan::math::check_consistent_sizes_mvt ( const char *  , const char *  , const T1 &    )

inline

Base case of recursion, this function is a no-op.

Template Parameters

T1	type of first input

Definition at line 22 of file check_consistent_sizes_mvt.hpp.

template<typename T1 , typename T2 , typename... Ts>

void stan::math::check_consistent_sizes_mvt ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2, const Ts &...  names_and_xs  )

inline

Check that the provided multivariate inputs are of consistent size with each other.

Two multivariate inputs are of consistent size if both are std::vectors of the same size, or if at least one is a not an std::vector.

E.g.: check_consistent_sizes_mvt("some_function", "x1", x1, "x2", x2, etc.).

Template Parameters

T1	type of first input
T2	type of second input
Ts	type of other inputs

Parameters

function	function name (for error messages)
name1	name of variable corresponding to first input
x1	first input
name2	name of variable corresponding to second input
x2	second input
names_and_xs	more inputs

Exceptions

<tt>invalid_argument</tt>

if sizes are inconsistent

Definition at line 46 of file check_consistent_sizes_mvt.hpp.

template<typename Mat , require_matrix_t< Mat > * = nullptr>

void stan::math::check_corr_matrix ( const char *  function, const char *  name, const Mat &  y  )

inline

Throw an exception if the specified matrix is not a valid correlation matrix.

A valid correlation matrix is symmetric positive definite, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive).

Template Parameters

Mat	Type inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Name of the function this was called from
name	Name of the variable
y	Matrix to test

Exceptions

<tt>std::invalid_argument</tt>	if the matrix is not square
<tt>std::domain_error</tt>	if the matrix is non-symmetric, diagonals not near 1, not positive definite, or any of the elements are NaN

Definition at line 34 of file check_corr_matrix.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_corr_matrix	(	const char *	function,
		const char *	name,
		const StdVec &	y
	)

Throw an exception if the specified matrix is not a valid correlation matrix.

A valid correlation matrix is symmetric positive definite, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive).

Template Parameters

StdVec

A standard vector with inner type either inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1.

Parameters

function	Name of the function this was called from
name	Name of the variable
y	Matrix to test

Exceptions

<tt>std::invalid_argument</tt>	if the matrix is not square
<tt>std::domain_error</tt>	if the matrix is non-symmetric, diagonals not near 1, not positive definite, or any of the elements are NaN

Definition at line 76 of file check_corr_matrix.hpp.

template<typename Mat , require_matrix_t< Mat > * = nullptr>

void stan::math::check_cov_matrix ( const char *  function, const char *  name, const Mat &  y  )

inline

Throw an exception if the specified matrix is not a valid covariance matrix.

A valid covariance matrix is a square, symmetric matrix that is positive definite.

Template Parameters

Mat	Type inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<tt>std::invalid_argument</tt>	if the matrix is not square or if the matrix is 0x0
<tt>std::domain_error</tt>	if the matrix is not symmetric, if the matrix is not positive definite, or if any element of the matrix is NaN

Definition at line 29 of file check_cov_matrix.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_cov_matrix	(	const char *	function,
		const char *	name,
		const StdVec &	y
	)

Throw an exception if the specified matrix is not a valid covariance matrix.

A valid covariance matrix is a square, symmetric matrix that is positive definite.

Template Parameters

StdVec

A standard vector with inner type either inheriting from MatrixBase with neither rows or columns defined at compile time to be equal to 1 or a var_value with the var's inner type inheriting from Eigen::MatrixBase with neither rows or columns defined at compile time to be equal to 1

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	standard vector of matrices to test.

Exceptions

<tt>std::invalid_argument</tt>	if the matrix is not square or if the matrix is 0x0
<tt>std::domain_error</tt>	if the matrix is not symmetric, if the matrix is not positive definite, or if any element of the matrix is NaN

Definition at line 52 of file check_cov_matrix.hpp.

template<typename T_y >

void stan::math::check_finite ( const char *  function, const char *  name, const T_y &  y  )

inline

Return true if all values in y are finite.

y can be a scalar, std::vector or Eigen type.

Template Parameters

T_y

type of y

Parameters

function	name of function (for error messages)
name	variable name (for error messages)
y	scalar or container to test

Returns

true if all values are finite

Definition at line 28 of file check_finite.hpp.

void stan::math::check_flag_kinsol ( int  flag, long int  max_num_steps  )

inline

Throws an exception message when the function KINSol() (call to the solver) fails.

When the exception is caused by a tuning parameter the user controls, gives a specific error.

Parameters

flag	Error flag
max_num_steps	Maximum number of iterations the algebra solver should take before throwing an error

Exceptions

<code>std::domain_error</code>	if flag means maximum number of iterations exceeded in the algebra solver.
<code>std::runtime_error</code>	if the flag is negative for any other reason.

Definition at line 40 of file check_flag_sundials.hpp.

void stan::math::check_flag_sundials ( int  flag, const char *  func_name  )

inline

Throws a std::runtime_error exception when a Sundial function fails (i.e.

returns a negative flag)

Parameters

flag	Error flag
func_name	Name of the function that returned the flag

Exceptions

<code>std::runtime_error</code>

if the flag is negative

Definition at line 18 of file check_flag_sundials.hpp.

template<typename T_y , typename T_low , require_all_stan_scalar_t< T_y, T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not strictly greater than low.

Throw an exception if each element of y is not strictly greater than the associated element in low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A scalar type type
T_low	A scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_low	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 36 of file check_greater.hpp.

template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_low > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not strictly greater than each element of low.

Throw an exception if each element of y is not strictly greater than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A scalar type
T_low	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar
T_low	A scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 72 of file check_greater.hpp.

template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not strictly greater than each element of low.

Throw an exception if each element of y is not strictly greater than the associated element in low.

Throw an exception if each element of y is not strictly greater than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A scalar type
T_low	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_low	A scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
T_low	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
T_low	A scalar type or the same type as the underlying type in T_y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A scalar type or the same type as the inner type of T_low
T_low	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

Idxs	A parameter pack of Integral types
T_y	A standard vector type whose value_type is a scalar, type inheriting from Eigen::EigenBase, or another standard vector
T_low	A standard vector type whose value_type is a scalar, type inheriting from Eigen::EigenBase, or another standard vector

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 108 of file check_greater.hpp.

template<typename T_y , typename T_low , require_all_stan_scalar_t< T_y, T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not greater or equal than low.

Throw an exception if each element of y is not greater or equal than the associated element in low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A scalar type
T_low	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_low	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 36 of file check_greater_or_equal.hpp.

template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_low > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not greater or equal than each element of low.

Throw an exception if each element of y is not greater or equal than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A scalar type
T_low	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar
T_low	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 74 of file check_greater_or_equal.hpp.

template<typename T_y , typename T_low , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_low > * = nullptr, typename... Idxs>

void stan::math::check_greater_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, Idxs...  idxs  )

inline

Throw an exception if y is not greater or equal than each element of low.

Throw an exception if each element of y is not greater or equal than the associated element in low.

Throw an exception if each element of y is not greater or equal than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A scalar type
T_low	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_low	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar
T_low	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
T_low	A standard vector type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A standard vector
T_low	A standard vector type with a value_type of a standard
Idxs	A parameter pack of Integral types vector or type inheriting from Eigen::DenseBase

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	A standard vector type
T_low	A standard vector type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not greater or equal to low or if any element of y or low is NaN

Definition at line 112 of file check_greater_or_equal.hpp.

template<typename T >

void stan::math::check_ldlt_factor ( const char *  function, const char *  name, LDLT_factor< T > &  A  )

inline

Raise domain error if the specified LDLT factor is invalid.

An LDLT_factor is invalid if it was constructed from a matrix that is not positive definite.

Template Parameters

T	Type matrix of LDLT

Parameters

[in]	function	name of function for error messages
[in]	name	variable name for error messages
[in]	A	the LDLT factor to check for validity

Exceptions

<code>std::domain_error</code>

if the LDLT factor is invalid

Definition at line 25 of file check_ldlt_factor.hpp.

template<typename T_y , typename T_high , require_all_stan_scalar_t< T_y, T_high > * = nullptr, typename... Idxs>

void stan::math::check_less ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not strictly less than high.

Throw an exception if each element of y is not strictly less than each element of high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_high	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 35 of file check_less.hpp.

template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_high > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_high > * = nullptr, typename... Idxs>

void stan::math::check_less ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not strictly less than each element of high.

Throw an exception if each element of y is not strictly less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 71 of file check_less.hpp.

template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_high > * = nullptr, typename... Idxs>

void stan::math::check_less ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not strictly less than each element of high.

Throw an exception if each element of y is not strictly less than each associated element of high.

Throw an exception if each element of y is not strictly less than each element of high.

Throw an exception if each element of y is not strictly less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
T_high	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
T_high	A scalar type or the same type as the inner type of T_high
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type or the same type as the inner type of T_high
T_high	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector type whose value_type is a scalar, type inheriting from Eigen::DenseBase, or another standard vector
T_high	A standard vector type whose value_type is a scalar, type inheriting from Eigen::DenseBase, or another standard vector
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 107 of file check_less.hpp.

template<typename T_y , typename T_high , require_all_stan_scalar_t< T_y, T_high > * = nullptr, typename... Idxs>

void stan::math::check_less_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not less than high.

Throw an exception if each element of y is not less than the associated element of high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_high	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 36 of file check_less_or_equal.hpp.

template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_vector_t< T_high > * = nullptr, require_not_std_vector_vt< is_container_or_var_matrix, T_high > * = nullptr, typename... Idxs>

void stan::math::check_less_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not less than each element of high.

Throw an exception if each element of y is not less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 74 of file check_less_or_equal.hpp.

template<typename T_y , typename T_high , require_stan_scalar_t< T_y > * = nullptr, require_dense_dynamic_t< T_high > * = nullptr, typename... Idxs>

void stan::math::check_less_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_high &  high, Idxs...  idxs  )

inline

Throw an exception if y is not less than each element of high.

Throw an exception if each element of y is not less than each associated element of high.

Throw an exception if each element of y is not less than the associated element of high.

Throw an exception if each element of y is not less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type
T_high	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type inheriting from Eigen::DenseBase or a var_value with the var's inner type inheriting from Eigen::DenseBase where the compile time number of rows or columns is not equal to one
T_high	A scalar type
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
T_high	A standard vector or type inheriting from Eigen::DenseBase with compile time rows or columns equal to one and value_type equal to a stan scalar.
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
T_high	A scalar or the same value_type of T_y
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A scalar type or the same value_type of T_high
T_high	A standard vector type with a value_type of a standard vector or type inheriting from Eigen::DenseBase
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	A standard vector type whose value_type is a scalar, type inheriting from Eigen::EigenBase, or another standard vector
T_high	A standard vector type whose value_type is a scalar, type inheriting from Eigen::EigenBase, or another standard vector
Idxs	A parameter pack of Integral types

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound
idxs	Pack of integral types to construct lazily construct the error message indices

Exceptions

<tt>std::domain_error</tt>

if y is not less than high or if any element of y or high is NaN

Definition at line 112 of file check_less_or_equal.hpp.

template<typename T_y , require_eigen_t< T_y > * = nullptr>

void stan::math::check_lower_triangular ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if the specified matrix is lower triangular.

A matrix x is not lower triangular if there is a non-zero entry x[m, n] with m < n. This function only inspects the upper triangular portion of the matrix, not including the diagonal.

Template Parameters

T	Type of the matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::domain_error</code>

if the matrix is not lower triangular or if any element in the upper triangular portion is NaN

Definition at line 27 of file check_lower_triangular.hpp.

template<typename T1 , typename T2 , require_all_not_matrix_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

void stan::math::check_matching_dims ( const char *  function, const char *  name1, const T1 &  y1, const char *  name2, const T2 &  y2  )

inline

Check if the two containers have the same dimensions.

Check if two matrices have the same row and column dimensions.

Template Parameters

T1	type of the first container
T2	type of the second container

Parameters

function	name of function (for error messages)
name1	variable name for the first container (for error messages)
y1	first container to test
name2	variable name for the second container (for error messages)
y2	second container to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the containers do not match

Template Parameters

T1	Either an Eigen type or a var_value with underlying Eigen type.
T2	Either an Eigen type or a var_value with underlying Eigen type.

Parameters

function	name of function (for error messages)
name1	variable name for the first container (for error messages)
y1	first matrix to test
name2	variable name for the second container (for error messages)
y2	second matrix to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the containers do not match

Template Parameters

T1	Either an Eigen type, a var_value with underlying Eigen type, or scalar.
T2	Either an Eigen type, a var_value with underlying Eigen type, or scalar.

Parameters

function	name of function (for error messages)
name1	variable name for the first container (for error messages)
y1	first argument to test
name2	variable name for the second container (for error messages)
y2	second argument to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the containers do not match

Definition at line 29 of file check_matching_dims.hpp.

template<bool check_compile, typename Mat1 , typename Mat2 , typename = require_all_eigen_t<Mat1, Mat2>>

void stan::math::check_matching_dims ( const char *  function, const char *  name1, const Mat1 &  y1, const char *  name2, const Mat2 &  y2  )

inline

Check if the two matrices are of the same size.

This function checks the runtime sizes and can also check the static sizes as well. For example, a 4x1 matrix is not the same as a vector with 4 elements.

Template Parameters

check_compile	Whether to check the static sizes
Mat1	type of the first matrix
Mat2	type of the second matrix

Parameters

function	name of function (for error messages)
name1	variable name for the first matrix (for error messages)
y1	first matrix to test
name2	variable name for the second matrix (for error messages)
y2	second matrix to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the matrices do not match

Definition at line 138 of file check_matching_dims.hpp.

template<typename T_y1 , typename T_y2 >

void stan::math::check_matching_sizes ( const char *  function, const char *  name1, const T_y1 &  y1, const char *  name2, const T_y2 &  y2  )

inline

Check if two structures at the same size.

This function only checks the runtime sizes for variables that implement a size() method.

Template Parameters

T_y1	Type of the first variable
T_y2	Type of the second variable

Parameters

function	Function name (for error messages)
name1	First variable name (for error messages)
y1	First variable
name2	Second variable name (for error messages)
y2	Second variable

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 24 of file check_matching_sizes.hpp.

template<typename T1 , typename T2 >

void stan::math::check_multiplicable ( const char *  function, const char *  name1, const T1 &  y1, const char *  name2, const T2 &  y2  )

inline

Check if the matrices can be multiplied.

This checks the runtime sizes to determine whether the two matrices are multiplicable. This allows Eigen matrices, vectors, and row vectors to be checked.

Template Parameters

T1	Type of first matrix
T2	Type of second matrix

Parameters

function	Function name (for error messages)
name1	Variable name for the first matrix (for error messages)
y1	First matrix to test, requires class access to .rows() and .cols()
name2	Variable name for the second matrix (for error messages)
y2	Second matrix to test, requires class access to .rows() and .cols()

Exceptions

<code>std::invalid_argument</code>

if the matrices are not multiplicable or if either matrix is size 0 for either rows or columns

Definition at line 30 of file check_multiplicable.hpp.

template<typename T_y >

void stan::math::check_nonnegative ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is non-negative.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if y is negative or if any element of y is NaN.

Definition at line 24 of file check_nonnegative.hpp.

template<typename T_y >

void stan::math::check_nonzero_size ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if the specified matrix/vector is of non-zero size.

Throws a std:invalid_argument otherwise. The message will indicate that the variable name "has size 0".

Template Parameters

T_y	Type of container

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Container to test. This will accept matrices and vectors

Exceptions

<code>std::invalid_argument</code>

if the specified matrix/vector has zero size

Definition at line 22 of file check_nonzero_size.hpp.

template<typename T_y >

void stan::math::check_not_nan ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is not NaN.

This function is vectorized and will check each element of y. If any element is NaN, this function will throw an exception.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if any element of y is NaN

Definition at line 26 of file check_not_nan.hpp.

template<typename T_y , require_vector_t< T_y > * = nullptr, require_not_std_vector_t< T_y > * = nullptr>

void stan::math::check_ordered	(	const char *	function,
		const char *	name,
		const T_y &	y
	)

Throw an exception if the specified vector is not sorted into strictly increasing order.

Throw an exception if each vector in a standard vector is not sorted into strictly increasing order.

Template Parameters

T_y	A type inheriting from EigenBase with either 1 compile time row or column

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<tt>std::domain_error</tt>

if the vector elements are not ordered, if there are duplicated values, or if any element is NaN

Template Parameters

T_y	A standard vector with an inner vector type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test

Exceptions

<tt>std::domain_error</tt>

if the vector elements are not ordered, if there are duplicated values, or if any element is NaN

Definition at line 30 of file check_ordered.hpp.

template<typename T_y , require_std_vector_vt< is_stan_scalar, T_y > * = nullptr>

void stan::math::check_ordered	(	const char *	function,
		const char *	name,
		const T_y &	y
	)

Throw an exception if the specified vector is not sorted into strictly increasing order.

Template Parameters

T_y	A standard vector with inner scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test

Exceptions

<tt>std::domain_error</tt>

if the vector elements are not ordered, if there are duplicated values, or if any element is NaN

Definition at line 61 of file check_ordered.hpp.

template<typename EigMat , require_matrix_t< EigMat > * = nullptr>

void stan::math::check_pos_definite ( const char *  function, const char *  name, const EigMat &  y  )

inline

Check if the specified square, symmetric matrix is positive definite.

This computes an LDLT decomposition to establish positive definiteness, so it can be expensive. If an LDLT or LLT decomposition is available, that should be tested instead.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

function	function name (for error messages)
name	variable name (for error messages)
y	matrix to test

Exceptions

std::invalid_argument	if the matrix is not square or if the matrix has 0 size.
std::domain_error	if the matrix is not symmetric, if it is not positive definite, or if any element is NaN

Definition at line 34 of file check_pos_definite.hpp.

template<typename Derived >

void stan::math::check_pos_definite ( const char *  function, const char *  name, const Eigen::LDLT< Derived > &  cholesky  )

inline

Check if the specified LDLT decomposition of a matrix is positive definite.

Template Parameters

Derived

type of the Eigen::LDLT decomposition

Parameters

function	function name (for error messages)
name	variable name (for error messages)
cholesky	Eigen::LDLT to test, whose progenitor must not have any NaN elements

Exceptions

std::domain_error

if the matrix is not positive definite

Definition at line 63 of file check_pos_definite.hpp.

template<typename Derived >

void stan::math::check_pos_definite ( const char *  function, const char *  name, const Eigen::LLT< Derived > &  cholesky  )

inline

Check if the specified LLT decomposition was successful.

Template Parameters

Derived

type of the Eigen::LLT decomposition

Parameters

function	function name (for error messages)
name	variable name (for error messages)
cholesky	Eigen::LLT to test, whose progenitor must not have any NaN elements

Exceptions

std::domain_error

if the decomposition failed or the diagonal of the L matrix is not positive

Definition at line 83 of file check_pos_definite.hpp.

template<typename EigMat , require_matrix_t< EigMat > * = nullptr>

void stan::math::check_pos_semidefinite ( const char *  function, const char *  name, const EigMat &  y  )

inline

Check if the specified matrix is positive definite.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square or if the matrix has 0 size.
<code>std::domain_error</code>	if the matrix is not symmetric, or if it is not positive semi-definite, or if any element of the matrix is NaN.

Definition at line 30 of file check_pos_semidefinite.hpp.

template<typename Derived >

void stan::math::check_pos_semidefinite ( const char *  function, const char *  name, const Eigen::LDLT< Derived > &  cholesky  )

inline

Check if the specified matrix is positive semidefinite.

Template Parameters

Derived

Derived type of the Eigen::LDLT transform.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
cholesky	Eigen::LDLT to test

Exceptions

<code>std::domain_error</code>

if the matrix is not positive semi-definite.

Definition at line 62 of file check_pos_semidefinite.hpp.

template<typename T_y >

void stan::math::check_positive ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is positive.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if y is negative or zero or if any element of y is NaN

Definition at line 27 of file check_positive.hpp.

void stan::math::check_positive ( const char *  function, const char *  name, const char *  expr, int  size  )

inline

Check if size is positive.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
expr	Expression for the dimension size (for error messages)
size	Size value to check

Exceptions

<code>std::invalid_argument</code>

if size is zero or negative.

Definition at line 42 of file check_positive.hpp.

template<typename T_y >

void stan::math::check_positive_finite ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is positive and finite.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if any element of y is not positive or if any element of y is NaN.

Definition at line 22 of file check_positive_finite.hpp.

template<typename Vec , require_vector_t< Vec > * = nullptr, require_not_std_vector_t< Vec > * = nullptr>

void stan::math::check_positive_ordered	(	const char *	function,
		const char *	name,
		const Vec &	y
	)

Throw an exception if the specified the vector contains negative values or is not sorted into strictly increasing order.

Template Parameters

Vec	A type derived from Eigen::EigenBase with 1 compile time row or column

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<tt>std::domain_error</tt>

if the vector contains non-positive values, if the values are not ordered, if there are duplicated values, or if any element is NaN

Definition at line 31 of file check_positive_ordered.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_positive_ordered	(	const char *	function,
		const char *	name,
		const StdVec &	y
	)

Throw an exception if any of the vectors in a standard vector contains negative values or is not sorted into strictly increasing order.

Template Parameters

StdVec

A standard vector type with an value_type inheriting from Eigen::EigenBase with 1 compile time row or column

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<tt>std::domain_error</tt>

if the vector contains non-positive values, if the values are not ordered, if there are duplicated values, or if any element is NaN

Definition at line 63 of file check_positive_ordered.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index, int  nested_level, const char *  error_msg  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check
nested_level	Nested level (for error messages)
error_msg	Additional error message (for error messages)

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 24 of file check_range.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index, const char *  error_msg  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check
error_msg	Additional error message (for error messages)

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 49 of file check_range.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 68 of file check_range.hpp.

template<typename T_y , typename = require_matrix_t<T_y>>

void stan::math::check_row_index ( const char *  function, const char *  name, const T_y &  y, size_t  i  )

inline

Check if the specified index is a valid row of the matrix This check is 1-indexed by default.

This behavior can be changed by setting stan::error_index::value.

Template Parameters

T	Matrix type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	matrix to test
i	row index to check

Exceptions

<code>std::out_of_range</code>

if the index is out of range.

Definition at line 25 of file check_row_index.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

void stan::math::check_simplex	(	const char *	function,
		const char *	name,
		const T &	theta
	)

Throw an exception if the specified vector is not a simplex.

Throw an exception if each vector in a standard vector is not a simplex.

To be a simplex, all values must be greater than or equal to 0 and the values must sum to 1. A valid simplex is one where the sum of the elements is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

T	A type inheriting from Eigen::EigenBase

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test

Exceptions

<tt>std::invalid_argument</tt>	if theta is a 0-vector
<tt>std::domain_error</tt>	if the vector is not a simplex or if any element is NaN

To be a simplex, all values must be greater than or equal to 0 and the values must sum to 1. A valid simplex is one where the sum of the elements is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

T	A standard vector with inner type inheriting from Eigen::EigenBase

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test.

Exceptions

<tt>std::invalid_argument</tt>	if theta is a 0-vector
<tt>std::domain_error</tt>	if the vector is not a simplex or if any element is NaN

Definition at line 34 of file check_simplex.hpp.

template<typename T_size1 , typename T_size2 >

void stan::math::check_size_match ( const char *  function, const char *  name_i, T_size1  i, const char *  name_j, T_size2  j  )

inline

Check if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

function	Function name (for error messages)
name_i	Variable name 1 (for error messages)
i	Variable size 1
name_j	Variable name 2 (for error messages)
j	Variable size 2

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 24 of file check_size_match.hpp.

template<typename T_size1 , typename T_size2 >

void stan::math::check_size_match ( const char *  function, const char *  expr_i, const char *  name_i, T_size1  i, const char *  expr_j, const char *  name_j, T_size2  j  )

inline

Check if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

function	Function name (for error messages)
expr_i	Expression for variable name 1 (for error messages)
name_i	Variable name 1 (for error messages)
i	Variable size 1
expr_j	Expression for variable name 2 (for error messages)
name_j	Variable name 2 (for error messages)
j	Variable size 2

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 50 of file check_size_match.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

void stan::math::check_sorted	(	const char *	function,
		const char *	name,
		const EigVec &	y
	)

Check if the specified vector is sorted into increasing order (repeated values are okay).

Template Parameters

EigVec

A type derived from EigenBase with 1 compile time row or column.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<code>std::domain_error</code>

if the vector elements are not sorted, or if any element is NaN.

Definition at line 26 of file check_sorted.hpp.

template<typename T_y >

void stan::math::check_sorted	(	const char *	function,
		const char *	name,
		const std::vector< T_y > &	y
	)

Check if the specified vector is sorted into increasing order (repeated values are okay).

Template Parameters

T_y	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test

Exceptions

<code>std::domain_error</code>

if the vector elements are not sorted, or if any element is NaN.

Definition at line 56 of file check_sorted.hpp.

template<typename T_y , require_any_t< is_matrix< T_y >, is_prim_or_rev_kernel_expression< T_y >> * = nullptr>

void stan::math::check_square ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if the specified matrix is square.

This check allows 0x0 matrices.

Template Parameters

T	Type of matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>

if the matrix is not square

Definition at line 23 of file check_square.hpp.

template<typename T >

void stan::math::check_std_vector_index ( const char *  function, const char *  name, const std::vector< T > &  y, int  i  )

inline

Check if the specified index is valid in std vector This check is 1-indexed by default.

This behavior can be changed by setting stan::error_index::value.

Template Parameters

T	Scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test
i	Index

Exceptions

<code>std::out_of_range</code>

if the index is out of range.

Definition at line 26 of file check_std_vector_index.hpp.

template<typename EigMat , require_matrix_t< EigMat > * = nullptr>

void stan::math::check_symmetric ( const char *  function, const char *  name, const EigMat &  y  )

inline

Check if the specified matrix is symmetric.

The error message is either 0 or 1 indexed, specified by stan::error_index::value.

Template Parameters

EigMat

Type of matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square.
<code>std::domain_error</code>	if any element not on the main diagonal is NaN

Definition at line 32 of file check_symmetric.hpp.

template<typename Vec , require_vector_t< Vec > * = nullptr, require_not_std_vector_t< Vec > * = nullptr>

void stan::math::check_unit_vector	(	const char *	function,
		const char *	name,
		const Vec &	theta
	)

Throw an exception if the specified vector does not have unit Euclidiean length.

A valid unit vector is one where the square of the elements summed is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

Vec	A type derived from Eigen::EigenBase with either dynamic rows or columns but not both

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test

Exceptions

<tt>std::invalid_argument</tt>	if theta is a 0-vector
<tt>std::domain_error</tt>	if the vector is not a unit vector or if any element is NaN

Definition at line 37 of file check_unit_vector.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr>

void stan::math::check_unit_vector	(	const char *	function,
		const char *	name,
		const StdVec &	theta
	)

Throw an exception if the each element in a standard vector does not have unit Euclidiean length.

Template Parameters

StdVec

A standard vector with inner type inheriting from Eigen::EigenBase

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test

Exceptions

<tt>std::invalid_argument</tt>	if theta is a 0-vector
<tt>std::domain_error</tt>	if the vector is not a unit vector or if any element is NaN

Definition at line 67 of file check_unit_vector.hpp.

template<typename Mat , require_any_t< is_matrix< Mat >, is_prim_or_rev_kernel_expression< Mat >> * = nullptr>

void stan::math::check_vector ( const char *  function, const char *  name, const Mat &  x  )

inline

Check the input is either a row vector or column vector or a matrix with a single row or column.

Template Parameters

Mat	Input type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
x	Input

Exceptions

<code>std::invalid_argument</code>

if x is not a row or column vector.

Definition at line 32 of file check_vector.hpp.

template<typename T , require_any_t< is_vector< T >, is_prim_or_rev_kernel_expression< T >> * = nullptr>

void stan::math::check_vector_index ( const char *  function, const char *  name, const T &  y, size_t  i  )

inline

Check if the specified index is a valid element of the row or column vector This check is 1-indexed by default.

This behavior can be changed by setting stan::error_index::value.

Template Parameters

T	Vector type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	vector to test
i	row index to check

Exceptions

<code>std::out_of_range</code>

if the index is out of range.

Definition at line 27 of file check_vector_index.hpp.

template<typename T_y , typename T_dof >

return_type_t<T_y, T_dof> stan::math::chi_square_lpdf ( const T_y &  y, const T_dof &  nu  )

inline

Definition at line 102 of file chi_square_lpdf.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

plain_type_t<T> stan::math::chol2inv

(

const T &

L

)

Returns the inverse of the matrix whose Cholesky factor is L.

Template Parameters

T	type of elements in the matrix

Parameters

L	Matrix that is a Cholesky factor.

Returns

The matrix inverse of L * L'

Exceptions

std::domain_error

If the input matrix is not square or lower triangular

Definition at line 24 of file chol2inv.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>

Eigen::Matrix<value_type_t<EigVec>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain ( const EigVec &  y, int  K  )

inline

Definition at line 17 of file cholesky_corr_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cholesky_corr_constrain	(	const T &	y,
		int	K
	)

Return the Cholesky factor of the correlation matrix of the sepcified size read from the unconstrained vector y.

A total of K choose 2 elements are required to build a K by K Cholesky factor.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

y	Vector of unconstrained values
K	number of rows

Returns

Cholesky factor of correlation matrix

Definition at line 28 of file cholesky_corr_constrain.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

Eigen::Matrix<value_type_t<EigVec>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain ( const EigVec &  y, int  K, return_type_t< EigVec > &  lp  )

inline

Definition at line 48 of file cholesky_corr_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cholesky_corr_constrain	(	const T &	y,
		int	K,
		scalar_type_t< T > &	lp
	)

Return the Cholesky factor of the correlation matrix of the sepcified size read from the unconstrained vector y.

A total of K choose 2 elements are required to build a K by K Cholesky factor.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

	y	Vector of unconstrained values
	K	number of rows
[out]	lp	Log density that is incremented with the log Jacobian

Returns

Cholesky factor of correlation matrix

Definition at line 92 of file cholesky_corr_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::cholesky_corr_constrain ( const T &  y, int  K, return_type_t< T > &  lp  )

inline

Return The cholesky of a KxK correlation matrix.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	y	Linearly Serialized vector of size (K * (K - 1))/2 holding the column major order elements of the lower triangurlar
	K	The size of the matrix to return
[in,out]	lp	log density accumulator

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	y	Linearly Serialized vector of size (K * (K - 1))/2 holding the column major order elements of the lower triangurlar
	K	The size of the matrix to return
[in,out]	lp	log density accumulator

Definition at line 94 of file cholesky_corr_constrain.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

auto stan::math::cholesky_corr_free

(

const T &

x

)

Overload of cholesky_corr_free() to untransform each matrix in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase.

Parameters

x	The standard vector to untransform.

Definition at line 14 of file cholesky_corr_free.hpp.

matrix_cl<double> stan::math::cholesky_decompose ( const matrix_cl< double > &  A )

inline

Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix on the OpenCL device.

The return value \(L\) will be a lower-triangular matrix such that the original matrix \(A\) is given by

\(A = L \times L^T\).

Parameters

A	Input square matrix

Returns

Square root of matrix.

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 25 of file cholesky_decompose.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::cholesky_decompose ( const var_value< T > &  A )

inline

Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric reverse mode matrix on the OpenCL device.

The return value \(L\) will be a lower-triangular matrix such that the original matrix \(A\) is given by

\(A = L \times L^T\).

Parameters

A	Input square matrix

Returns

Square root of matrix.

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 26 of file cholesky_decompose.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vt< is_var, EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime> stan::math::cholesky_decompose ( const EigMat &  m )

inline

Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix.

The return value \(L\) will be a lower-triangular matrix such that the original matrix \(A\) is given by

\(A = L \times L^EigMat\).

Template Parameters

EigMat

type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

m	Symmetric matrix.

Returns

Square root of matrix.

Note

Because OpenCL only works on doubles there are two cholesky_decompose functions. One that works on doubles and another that works on all other types (this one).

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 33 of file cholesky_decompose.hpp.

template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>

auto stan::math::cholesky_decompose ( const EigMat &  A )

inline

Reverse mode specialization of cholesky decomposition.

Internally calls Eigen::LLT rather than using stan::math::cholesky_decompose in order to use an inplace decomposition.

Note chainable stack varis are created below in Matrix<var, -1, -1>

Parameters

A

Matrix

Returns

L cholesky factor of A

Definition at line 135 of file cholesky_decompose.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::cholesky_decompose ( const T &  A )

inline

Reverse mode specialization of Cholesky decomposition.

Internally calls Eigen::LLT rather than using stan::math::cholesky_decompose in order to use an inplace decomposition.

Note chainable stack varis are created below in Matrix<var, -1, -1>

Template Parameters

T	A var_value holding an inner eigen type.

Parameters

A	A square positive definite matrix with no nan values.

Returns

L Cholesky factor of A

Definition at line 171 of file cholesky_decompose.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cholesky_factor_constrain	(	const T &	x,
		int	M,
		int	N
	)

Return the Cholesky factor of the specified size read from the specified vector.

A total of (N choose 2) + N + (M - N) * N elements are required to read an M by N Cholesky factor.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	Vector of unconstrained values
M	number of rows
N	number of columns

Returns

Cholesky factor

Definition at line 30 of file cholesky_factor_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain ( const T &  x, int  M, int  N  )

inline

Return the Cholesky factor of the specified size read from the specified vector.

A total of (N choose 2) + N + (M - N) * N elements are required to read an M by N Cholesky factor.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Vector of unconstrained values
M	number of rows
N	number of columns

Returns

Cholesky factor

Definition at line 30 of file cholesky_factor_constrain.hpp.

template<typename T , require_eigen_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain ( const T &  x, int  M, int  N, return_type_t< T > &  lp  )

inline

Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log absolute Jacobian determinant adjustment of the transform.

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Vector of unconstrained values
M	number of rows
N	number of columns
lp	Log probability that is incremented with the log absolute Jacobian determinant

Returns

Cholesky factor

Definition at line 76 of file cholesky_factor_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cholesky_factor_constrain	(	const T &	x,
		int	M,
		int	N,
		scalar_type_t< T > &	lp
	)

Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform.

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

	x	Vector of unconstrained values
	M	number of rows
	N	number of columns
[out]	lp	Log density that is incremented with the log Jacobian

Returns

Cholesky factor

Definition at line 89 of file cholesky_factor_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::cholesky_factor_constrain ( const T &  x, int  M, int  N, return_type_t< T > &  lp  )

inline

Return the Cholesky factor of the specified size read from the specified vector.

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Vector of unconstrained values
	M	number of rows
	N	number of columns
[in,out]	lp	log density accumulator

Returns

Cholesky factor

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Vector of unconstrained values
	M	number of rows
	N	number of columns
[in,out]	lp	log density accumulator

Returns

Cholesky factor

Definition at line 110 of file cholesky_factor_constrain.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, 1> stan::math::cholesky_factor_free

(

const T &

y

)

Return the unconstrained vector of parameters corresponding to the specified Cholesky factor.

A Cholesky factor must be lower triangular and have positive diagonal elements.

Template Parameters

T	type of the Cholesky factor (must be derived from Eigen::MatrixBase)

Parameters

y	Cholesky factor.

Returns

Unconstrained parameters for Cholesky factor.

Exceptions

std::domain_error

If the matrix is not a Cholesky factor.

Definition at line 26 of file cholesky_factor_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::cholesky_factor_free

(

const T &

x

)

Overload of cholesky_factor_free() to untransform each matrix in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase.

Parameters

x	The standard vector to untransform.

Definition at line 59 of file cholesky_factor_free.hpp.

int stan::math::choose ( int  n, int  k  )

inline

Return the binomial coefficient for the specified integer arguments.

The binomial coefficient, \({n \choose k}\), read "n choose k", is defined for \(0 \leq k \leq n\) (otherwise return 0) by

\({n \choose k} = \frac{n!}{k! (n-k)!}\).

Parameters

n	total number of objects
k	number of objects chosen

Returns

n choose k or 0 iff k > n

Exceptions

std::domain_error

if either argument is negative or the result will not fit in an int type

Definition at line 29 of file choose.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::choose ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the binomial coefficient function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Binomial coefficient function applied to the two inputs.

Definition at line 52 of file choose.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::col ( const T &  m, size_t  j  )

inline

Return the specified column of the specified matrix using start-at-1 indexing.

This is equivalent to calling m.col(i - 1) and assigning the resulting template expression to a column vector.

Template Parameters

T	type of the matrix

Parameters

m	Matrix.
j	Column index (count from 1).

Returns

Specified column of the matrix.

Exceptions

std::out_of_range

if j is out of range.

Definition at line 24 of file col.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

Eigen::Index stan::math::cols ( const T &  m )

inline

Return the number of columns in the specified matrix, vector, or row vector.

Template Parameters

T	type of the matrix

Parameters

[in]

m

Input matrix, vector, or row vector.

Returns

Number of columns.

Definition at line 19 of file cols.hpp.

template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::columns_dot_product ( const T_a &  a, const T_b &  b  )

inline

Returns the dot product of columns of the specified matrices.

Template Parameters

T_a	type of the first matrix
T_b	type of the second matrix

Parameters

a	Matrix of first vectors.
b	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the matrices are not the same size

Definition at line 27 of file columns_dot_product.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>

Eigen::Matrix< return_type_t< Mat1, Mat2 >, 1, Mat1::ColsAtCompileTime > stan::math::columns_dot_product ( const Mat1 &  v1, const Mat2 &  v2  )

inline

Returns the dot product of columns of the specified matrices.

Template Parameters

Mat1	type of the first matrix (must be derived from Eigen::MatrixBase)
Mat2	type of the second matrix (must be derived from Eigen::MatrixBase)

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 28 of file columns_dot_product.hpp.

template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::columns_dot_product ( T1 &&  v1, T2 &&  v2  )

inline

Returns the dot product of columns of the specified matrices.

Template Parameters

T1	type of the first matrix
T2	type of the second matrix

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::invalid_argument

If the vectors are not the same size

Definition at line 30 of file columns_dot_product.hpp.

template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>

auto stan::math::columns_dot_product ( const Mat1 &  v1, const Mat2 &  v2  )

inline

Returns the dot product of columns of the specified matrices.

This overload is used when at least one of Mat1 and Mat2 is a var_value<T> where T inherits from EigenBase. The other argument can be another var_value or a type that inherits from EigenBase.

Template Parameters

Mat1	type of the first matrix
Mat2	type of the second matrix

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 64 of file columns_dot_product.hpp.

template<typename Mat , require_eigen_vt< is_var, Mat > * = nullptr>

Eigen::Matrix<var, 1, Mat::ColsAtCompileTime> stan::math::columns_dot_self ( const Mat &  x )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

Mat	An Eigen matrix with a var scalar type.

Parameters

x

Matrix.

Definition at line 20 of file columns_dot_self.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_var, T > * = nullptr>

Eigen::Matrix<value_type_t<T>, 1, T::ColsAtCompileTime> stan::math::columns_dot_self ( const T &  x )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

x

Matrix.

Returns

Row vector containing the dot product of each column of the matrix with itself.

Definition at line 21 of file columns_dot_self.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::columns_dot_self ( const var_value< T > &  v )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

T	type of the matrix

Parameters

v

Matrix.

Definition at line 22 of file columns_dot_self.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

auto stan::math::columns_dot_self ( const T &  a )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

T	type of the matrix

Parameters

a

Matrix.

Returns

Row vector containing the dot product of each column of the matrix with itself.

Definition at line 24 of file columns_dot_self.hpp.

template<typename Mat , require_var_matrix_t< Mat > * = nullptr>

auto stan::math::columns_dot_self ( const Mat &  x )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

Mat	A var_value<> with an inner matrix type.

Parameters

x

Matrix.

Definition at line 36 of file columns_dot_self.hpp.

template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>

auto stan::math::colwise_optional_broadcast ( T &&  a )

inline

Broadcast an expression in colwise dimmension if the number of rows equals to 1.

In that case further expressions can use this expression as if had any number of rows, repeating the values.

Broadcasting evaluates argument expression multiple times. For performance reasons don't broadcast slow operations. Instead evaluate them in a separate kernel.

Template Parameters

T	type of input expression

Parameters

a	input expression

Returns

broadcast expression

Definition at line 240 of file optional_broadcast.hpp.

template<typename V >

std::complex<V> stan::math::complex_negate ( const std::complex< V > &  z )

inline

Return the complex negation of the specified complex argument.

Template Parameters

V	value type of complex argument

Parameters

[in]

z

argument

Returns

negation of argument

Definition at line 48 of file i_times.hpp.

std::complex<var> stan::math::conj ( const std::complex< var > &  z )

inline

Return the complex conjugate of the complex argument.

Parameters

[in]

z

argument

Returns

complex conjugate of the argument

Definition at line 17 of file conj.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::conj ( const std::complex< fvar< T >> &  z )

inline

Return the phase angle of the complex argument.

Template Parameters

T	value type of autodiff variable

Parameters

[in]

z

argument

Returns

phase angle of the argument

Definition at line 19 of file conj.hpp.

template<typename T , typename U >

T stan::math::copysign ( const T &  x, const U &  y  )

inline

Return the negation of the first argument if the first and second argument have different signs, otherwise return a copy of the first argument.

For the sake of this function, zero is considered positive. This function uses negation rather than literally copying signs to preserve derivatives.

Overload of std::copysign from cmath for argument-dependent lookup.

Template Parameters

T	type of first argument
U	type of second argument

Parameters

[in]	x	first complex argument
[in]	y	second complex argument

Returns

copy of the first argument, negated if necessary to match the sign of the second argument

Definition at line 30 of file copysign.hpp.

template<typename T , typename U >

std::complex<T> stan::math::copysign ( const std::complex< T > &  x, const std::complex< U > &  y  )

inline

Return the complex number composed of the real and complex parts with signs copied from the real and complex parts of the first arguments to the real and complex parts of the second.

This is an overload of the standard libary copysign for complex numbers that will be used with argument-dependent lookup. Rather than using the standard library copysign, it uses copysign_non_zero, which does not change sign if the reference value is zero (-0.0 or 0.0).

Template Parameters

T	value type of first argument
U	value type of second argument

Parameters

[in]	x	first complex argument
[in]	y	second complex argument

Returns

copy of second argument, with components negated if necessary to match sign of first argument

Definition at line 72 of file copysign.hpp.

template<typename T , typename U >

T stan::math::copysign_non_zero ( const T &  x, const U &  y  )

inline

Return the negation of the first argument if the first and second arguments have different signs and the first argument is not zero, otherwise return a copy of the first argument.

Template Parameters

T	type of first argument
U	type of second argument

Parameters

[in]	x	first complex argument
[in]	y	second complex argument

Returns

copy of the first argument, negated if necessary to match the sign of the second argument

See also

copysign

Definition at line 49 of file copysign.hpp.

template<typename T >

plain_type_t<T> stan::math::corr_constrain ( const T &  x )

inline

Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive).

The transform used is the hyperbolic tangent function,

\(f(x) = \tanh x = \frac{\exp(2x) - 1}{\exp(2x) + 1}\).

Template Parameters

T	type of value or container

Parameters

[in]

x

value or container

Returns

tanh transform

Definition at line 27 of file corr_constrain.hpp.

template<typename T_x , typename T_lp >

auto stan::math::corr_constrain ( const T_x &  x, T_lp &  lp  )

inline

Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive).

The transform used is as specified for corr_constrain(T). The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \tanh x | = \log (1 - \tanh^2 x)\).

Template Parameters

T_x	Type of scalar or container

Parameters

[in]	x	value or container
[in,out]	lp	log density accumulator

Definition at line 46 of file corr_constrain.hpp.

template<bool Jacobian, typename T_x , typename T_lp >

auto stan::math::corr_constrain ( const T_x &  x, T_lp &  lp  )

inline

Return the result of transforming the specified scalar or container of values to have a valid correlation value between -1 and 1 (inclusive).

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T_x	Type of scalar or container
T_lp	type of target log density accumulator

Parameters

[in]	x	value or container
[in,out]	lp	log density accumulator

Definition at line 68 of file corr_constrain.hpp.

template<typename T >

T stan::math::corr_free ( const T &  y )

inline

Return the unconstrained scalar that when transformed to a valid correlation produces the specified value.

This function inverts the transform defined for corr_constrain(T), which is the inverse hyperbolic tangent,

\( f^{-1}(y) = \mbox{atanh}\, y = \frac{1}{2} \log \frac{y + 1}{y - 1}\).

Template Parameters

T	Type of correlation

Parameters

[in]

y

correlation

Returns

free scalar that transforms to the specified input

Definition at line 28 of file corr_free.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::corr_matrix_constrain	(	const T &	x,
		Eigen::Index	k
	)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform based on partial correlations is as specified in

• Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100:1989–-2001.

The free vector entries are first constrained to be valid correlation values using corr_constrain(T).

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.

Exceptions

std::invalid_argument

if x is not a valid correlation matrix.

Definition at line 41 of file corr_matrix_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain ( const T &  x, Eigen::Index  k  )

inline

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform based on partial correlations is as specified in

• Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100:1989–-2001.

The free vector entries are first constrained to be valid correlation values using corr_constrain(T).

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.

Exceptions

std::invalid_argument

if x is not a valid correlation matrix.

Definition at line 41 of file corr_matrix_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::corr_matrix_constrain	(	const T &	x,
		Eigen::Index	k,
		scalar_type_t< T > &	lp
	)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform is as specified for corr_matrix_constrain(Matrix, size_t); the paper it cites also defines the Jacobians for correlation inputs, which are composed with the correlation constrained Jacobians defined in corr_constrain(T, double) for this function.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.
lp	Log probability reference to increment.

Definition at line 69 of file corr_matrix_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain ( const T &  x, Eigen::Index  k, return_type_t< T > &  lp  )

inline

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform is as specified for corr_matrix_constrain(Matrix, size_t); the paper it cites also defines the Jacobians for correlation inputs, which are composed with the correlation constrained Jacobians defined in corr_constrain(T, double) for this function.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.
lp	Log probability reference to increment.

Definition at line 70 of file corr_matrix_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::corr_matrix_constrain ( const T &  x, Eigen::Index  k, return_type_t< T > &  lp  )

inline

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Vector of unconstrained partial correlations
	k	Dimensionality of returned correlation matrix
[in,out]	lp	log density accumulator

Definition at line 97 of file corr_matrix_constrain.hpp.

template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::corr_matrix_constrain ( const T &  y, int  K, return_type_t< T > &  lp  )

inline

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Vector of unconstrained partial correlations
	k	Dimensionality of returned correlation matrix
[in,out]	lp	log density accumulator

Definition at line 126 of file corr_matrix_constrain.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, 1> stan::math::corr_matrix_free

(

const T &

y

)

Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed.

The constraining transform is defined as for corr_matrix_constrain(Matrix, size_t). The inverse transform in this function is simpler in that it only needs to compute the \(k \choose 2\) partial correlations and then free those.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

y	The correlation matrix to free.

Returns

Vector of unconstrained values that produce the specified correlation matrix when transformed.

Exceptions

std::domain_error	if the correlation matrix has no elements or is not a square matrix.
std::runtime_error	if the correlation matrix cannot be factorized by factor_cov_matrix() or if the sds returned by factor_cov_matrix() on log scale are unconstrained.

Definition at line 34 of file corr_matrix_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::corr_matrix_free

(

const T &

x

)

Overload of corr_matrix_free() to untransform each matrix in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase.

Parameters

x	The standard vector to untransform.

Definition at line 63 of file corr_matrix_free.hpp.

template<typename T >

fvar<T> stan::math::cos ( const fvar< T > &  x )

inline

Definition at line 14 of file cos.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::cos ( const var_value< T > &  A )

inline

Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise cos() of the input, in radians.

Definition at line 21 of file cos.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::cos ( const std::complex< fvar< T >> &  z )

inline

Return the cosine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

cosine of the argument

Definition at line 28 of file cos.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::cos ( const Container &  x )

inline

Returns the elementwise cos() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x	angles in radians

Returns

Cosine of each value in x.

Definition at line 44 of file cos.hpp.

var stan::math::cos ( var  a )

inline

Return the cosine of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \cos x = - \sin x\).

\[ \mbox{cos}(x) = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cos}(x)}{\partial x} = \begin{cases} -\sin(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Cosine of variable.

Definition at line 46 of file cos.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::cos ( const Container &  x )

inline

Version of cos() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Cosine of each value in x.

Definition at line 58 of file cos.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::cos ( const VarMat &  a )

inline

Return the cosine of a radian-scaled variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Variable for radians of angle.

Returns

Cosine of variable.

Definition at line 61 of file cos.hpp.

std::complex<var> stan::math::cos ( const std::complex< var > &  z )

inline

Return the cosine of the complex argument.

Parameters

[in]

z

argument

Returns

cosine of the argument

Definition at line 73 of file cos.hpp.

template<typename T >

fvar<T> stan::math::cosh ( const fvar< T > &  x )

inline

Definition at line 15 of file cosh.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::cosh ( const var_value< T > &  A )

inline

Returns the elementwise cosh() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise cosh() of the input, in radians.

Definition at line 21 of file cosh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::cosh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic cosine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic cosine of the argument

Definition at line 29 of file cosh.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::cosh ( const Container &  x )

inline

Returns the elementwise cosh() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x	angles in radians

Returns

Hyberbolic cosine of x.

Definition at line 43 of file cosh.hpp.

var stan::math::cosh ( const var &  a )

inline

Return the hyperbolic cosine of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \cosh x = \sinh x\).

\[ \mbox{cosh}(x) = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cosh}(x)}{\partial x} = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic cosine of variable.

Definition at line 44 of file cosh.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::cosh ( const Container &  x )

inline

Version of cosh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Hyberbolic cosine of x.

Definition at line 57 of file cosh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::cosh ( const VarMat &  a )

inline

Return the hyperbolic cosine of the specified variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a

Variable.

Returns

Hyperbolic cosine of variable.

Definition at line 58 of file cosh.hpp.

std::complex<var> stan::math::cosh ( const std::complex< var > &  z )

inline

Return the hyperbolic cosine of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic cosine of the argument

Definition at line 71 of file cosh.hpp.

template<typename... Pargs>

size_t stan::math::count_vars ( Pargs &&...  args )

inline

Count the number of vars in the input argument list.

Template Parameters

Pargs

Types of input arguments

Returns

Number of vars in input

Definition at line 146 of file count_vars.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 20 of file cov_exp_quad.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 31 of file cov_exp_quad.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 42 of file cov_exp_quad.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 53 of file cov_exp_quad.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const var &  sigma, const var &  l  )

inline

Deprecated:

use gp_exp_quad_cov_vari

Definition at line 154 of file cov_exp_quad.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, double  sigma, const var &  l  )

inline

Deprecated:

use gp_exp_quad_cov_vari

Definition at line 164 of file cov_exp_quad.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cov_matrix_constrain	(	const T &	x,
		Eigen::Index	K
	)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	The vector to convert to a covariance matrix.
K	The number of rows and columns of the resulting covariance matrix.

Exceptions

std::invalid_argument

if (x.size() != K + (K choose 2)).

Definition at line 32 of file cov_matrix_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain ( const T &  x, Eigen::Index  K  )

inline

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	The vector to convert to a covariance matrix.
K	The number of rows and columns of the resulting covariance matrix.

Exceptions

std::invalid_argument

if (x.size() != K + (K choose 2)).

Definition at line 32 of file cov_matrix_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain ( const T &  x, Eigen::Index  K, return_type_t< T > &  lp  )

inline

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	The vector to convert to a covariance matrix.
K	The dimensions of the resulting covariance matrix.
lp	Reference

Exceptions

std::domain_error

if (x.size() != K + (K choose 2)).

Definition at line 65 of file cov_matrix_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cov_matrix_constrain	(	const T &	x,
		Eigen::Index	K,
		scalar_type_t< T > &	lp
	)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	The vector to convert to a covariance matrix.
K	The dimensions of the resulting covariance matrix.
lp	Reference

Exceptions

std::domain_error

if (x.size() != K + (K choose 2)).

Definition at line 77 of file cov_matrix_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::cov_matrix_constrain ( const T &  x, Eigen::Index  K, return_type_t< T > &  lp  )

inline

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	The vector to convert to a covariance matrix
	K	The dimensions of the resulting covariance matrix
[in,out]	lp	log density accumulator

Exceptions

std::domain_error

if (x.size() != K + (K choose 2)).

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	The vector to convert to a covariance matrix
	K	The dimensions of the resulting covariance matrix
[in,out]	lp	log density accumulator

Exceptions

std::domain_error

if (x.size() != K + (K choose 2)).

Definition at line 108 of file cov_matrix_constrain.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cov_matrix_constrain_lkj	(	const T &	x,
		size_t	k
	)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values.

The input vector must be of length \(k \choose 2 + k\). The first \(k \choose 2\) values in the input represent unconstrained (partial) correlations and the last \(k\) are unconstrained standard deviations of the dimensions.

The transform scales the correlation matrix transform defined in corr_matrix_constrain(Matrix, size_t) with the constrained deviations.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 35 of file cov_matrix_constrain_lkj.hpp.

template<typename T , require_eigen_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj ( const T &  x, size_t  k  )

inline

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values.

The input vector must be of length \(k \choose 2 + k\). The first \(k \choose 2\) values in the input represent unconstrained (partial) correlations and the last \(k\) are unconstrained standard deviations of the dimensions.

The transform scales the correlation matrix transform defined in corr_matrix_constrain(Matrix, size_t) with the constrained deviations.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 35 of file cov_matrix_constrain_lkj.hpp.

template<typename T , require_eigen_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj ( const T &  x, size_t  k, return_type_t< T > &  lp  )

inline

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase and have one compile-time dimension equal to 1)

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.
lp	Log probability reference to increment.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 59 of file cov_matrix_constrain_lkj.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::cov_matrix_constrain_lkj	(	const T &	x,
		size_t	k,
		scalar_type_t< T > &	lp
	)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant.

The transform is defined as for cov_matrix_constrain(Matrix, size_t).

The log absolute Jacobian determinant is derived by composing the log absolute Jacobian determinant for the underlying correlation matrix as defined in cov_matrix_constrain(Matrix, size_t, T&) with the Jacobian of the transform of the correlation matrix into a covariance matrix by scaling by standard deviations.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.
lp	Log probability reference to increment.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 67 of file cov_matrix_constrain_lkj.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::cov_matrix_constrain_lkj ( const T &  x, size_t  k, return_type_t< T > &  lp  )

inline

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time rows or columns equal to 1

Parameters

	x	Input vector of unconstrained partial correlations and standard deviations
	k	Dimensionality of returned covariance matrix
[in,out]	lp	log density accumulator

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time rows or columns equal to 1

Parameters

	x	Input vector of unconstrained partial correlations and standard deviations
	k	Dimensionality of returned covariance matrix
[in,out]	lp	log density accumulator

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 87 of file cov_matrix_constrain_lkj.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, 1> stan::math::cov_matrix_free

(

const T &

y

)

The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2).

This is the inverse of the cov_matrix_constrain() function so that for any finite vector x of size K

• (K choose 2),

x == cov_matrix_free(cov_matrix_constrain(x, K)).

In order for this round-trip to work (and really for this function to work), the symmetric view of its lower-triangular view must be positive definite.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

y	Matrix of dimensions K by K such that he symmetric view of the lower-triangular view is positive definite.

Returns

Vector of size K plus (K choose 2) in (-inf, inf) that produces

Exceptions

std::domain_error

if y is not square, has zero dimensionality, or has a non-positive diagonal element.

Definition at line 38 of file cov_matrix_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::cov_matrix_free

(

const T &

x

)

Overload of cov_matrix_free() to untransform each matrix in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase.

Parameters

x	The standard vector to untransform.

Definition at line 69 of file cov_matrix_free.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, 1> stan::math::cov_matrix_free_lkj

(

const T &

y

)

Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix.

The constraining transform is defined as for cov_matrix_constrain(Matrix, size_t). The inverse first factors out the deviations, then applies the freeing transform of corr_matrix_free(Matrix&).

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

y	Covariance matrix to free.

Returns

Vector of unconstrained values that transforms to the specified covariance matrix.

Exceptions

std::domain_error	if the correlation matrix has no elements or is not a square matrix.
std::runtime_error	if the correlation matrix cannot be factorized by factor_cov_matrix()

Definition at line 31 of file cov_matrix_free_lkj.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::cov_matrix_free_lkj

(

const T &

x

)

Overload of cov_matrix_free_lkj() to untransform each matrix in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase.

Parameters

x	The standard vector to untransform.

Definition at line 60 of file cov_matrix_free_lkj.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

auto stan::math::crossprod ( const EigMat &  M )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Template Parameters

EigMat

type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

M	Matrix to multiply.

Returns

Transpose of M times M

Definition at line 20 of file crossprod.hpp.

template<typename T_A , typename = require_all_kernel_expressions_and_none_scalar_t<T_A>>

matrix_cl<typename std::decay_t<T_A>::Scalar> stan::math::crossprod ( T_A &&  A )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Template Parameters

T	type of elements in A

Parameters

A	input matrix

Returns

transpose(A) * A

Definition at line 21 of file crossprod.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::crossprod ( const var_value< T > &  M )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Template Parameters

T	Type of the matrix

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 22 of file crossprod.hpp.

template<typename T1 , typename T2 , require_all_not_rev_matrix_t< T1, T2 > * = nullptr>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> stan::math::csr_matrix_times_vector ( int  m, int  n, const T1 &  w, const std::vector< int > &  v, const std::vector< int > &  u, const T2 &  b  )

inline

Definition at line 75 of file csr_matrix_times_vector.hpp.

template<typename T1 , typename T2 , require_any_rev_matrix_t< T1, T2 > * = nullptr>

auto stan::math::csr_matrix_times_vector ( int  m, int  n, const T1 &  w, const std::vector< int > &  v, const std::vector< int > &  u, const T2 &  b  )

inline

Definition at line 78 of file csr_matrix_times_vector.hpp.

template<typename T >

std::vector<T> stan::math::cumulative_sum ( const std::vector< T > &  x )

inline

Return the cumulative sum of the specified vector.

The cumulative sum of a vector of values

 is the

\code x[0], x[1] + x[2], ..., x[1] + , ..., + x[x.size()-1] 

Template Parameters

T	type of elements in the vector

Parameters

x	Vector of values.

Returns

Cumulative sum of values.

Definition at line 25 of file cumulative_sum.hpp.

template<typename T_vec , require_all_kernel_expressions_and_none_scalar_t< T_vec > * = nullptr>

auto stan::math::cumulative_sum ( T_vec &&  v )

inline

Return the cumulative sum of the specified vector.

The cumulative sum of a vector of values

 is the

\code x[0], x[1] + x[2], ..., x[1] + , ..., + x[x.size()-1] 

Template Parameters

T	type of the vector

Parameters

x	Vector of values.

Returns

Cumulative sum of values.

Definition at line 25 of file cumulative_sum.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::cumulative_sum ( const var_value< T > &  A )

inline

Return the cumulative sum of the specified vector.

The cumulative sum of a vector of values

 is the

\code x[0], x[1] + x[2], ..., x[1] + , ..., + x[x.size()-1] 

Template Parameters

T	scalar type of the vector

Parameters

A	Vector of values

Returns

Cumulative sum of values

Definition at line 26 of file cumulative_sum.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

auto stan::math::cumulative_sum ( const EigVec &  m )

inline

Return the cumulative sum of the specified vector.

The cumulative sum is of the same type as the input and has values defined by

x(0), x(1) + x(2), ..., x(1) + , ..., + x(x.size()-1) 

Template Parameters

EigVec

type of the vector (must be derived from Eigen::MatrixBase and have one compile time dimension equal to 1)

Parameters

m	Vector of values.

Returns

Cumulative sum of values.

Definition at line 49 of file cumulative_sum.hpp.

void stan::math::cvodes_err_handler ( int  error_code, const char *  module, const char *  function, char *  msg, void *  eh_data  )

inline

Definition at line 13 of file cvodes_utils.hpp.

void stan::math::cvodes_set_options ( void *  cvodes_mem, long int  max_num_steps  )

inline

Definition at line 24 of file cvodes_utils.hpp.

template<typename Arith , typename = require_arithmetic_t<scalar_type_t<Arith>>>

Arith stan::math::deep_copy_vars ( Arith &&  arg )

inline

Forward arguments that do not contain vars.

There is no copying to be done.

Template Parameters

Arith

an arithmetic type.

Parameters

arg	For lvalue references this will be passed by reference. Otherwise it will be moved.

Definition at line 23 of file deep_copy_vars.hpp.

auto stan::math::deep_copy_vars ( const var &  arg )

inline

Copy the value of a var but reallocate a new vari.

Parameters

arg

A var

Returns

A new var

Definition at line 33 of file deep_copy_vars.hpp.

template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr>

auto stan::math::deep_copy_vars ( VarVec &&  arg )

inline

Copy the vars in arg but reallocate new varis for them.

Template Parameters

VarVec

A variant of std::vector<var>

Parameters

arg	A std::vector of vars

Returns

A new std::vector of vars

Definition at line 45 of file deep_copy_vars.hpp.

template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr>

auto stan::math::deep_copy_vars ( VecContainer &&  arg )

inline

Copy the vars in arg but reallocate new varis for them.

Template Parameters

VecContainer

std::vector<T> where T is another type containing vars

Parameters

arg	A std::vector of containers containing vars

Returns

A new std::vector of containers containing vars

Definition at line 63 of file deep_copy_vars.hpp.

template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr>

auto stan::math::deep_copy_vars ( EigT &&  arg )

inline

Copy the vars in arg but reallocate new varis for them.

Template Parameters

EigT	An Eigen type with var value type

Parameters

arg	An Eigen container of vars

Returns

A new Eigen container of vars

Definition at line 79 of file deep_copy_vars.hpp.

template<typename T , typename F >

void stan::math::derivative	(	const F &	f,
		const T &	x,
		T &	fx,
		T &	dfx_dx
	)

Return the derivative of the specified univariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

[in]	f	Function
[in]	x	Argument
[out]	fx	Value of function applied to argument
[out]	dfx_dx	Value of derivative

Definition at line 24 of file derivative.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

var stan::math::determinant ( const T &  m )

inline

Definition at line 13 of file determinant.hpp.

template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>

value_type_t<EigMat> stan::math::determinant ( const EigMat &  m )

inline

Definition at line 14 of file determinant.hpp.

template<typename T , require_eigen_vt< std::is_arithmetic, T > * = nullptr>

value_type_t<T> stan::math::determinant ( const T &  m )

inline

Returns the determinant of the specified square matrix.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

m	Specified matrix.

Returns

Determinant of the matrix.

Exceptions

std::domain_error

if matrix is not square.

Definition at line 21 of file determinant.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

Eigen::Matrix<value_type_t<EigVec>, Eigen::Dynamic, Eigen::Dynamic> stan::math::diag_matrix ( const EigVec &  v )

inline

Return a square diagonal matrix with the specified vector of coefficients as the diagonal values.

Template Parameters

EigVec

type of the vector (must be derived from Eigen::MatrixBase and have one compile time dimmension equal to 1)

Parameters

[in]

v

Specified vector.

Returns

Diagonal matrix with vector as diagonal values.

Definition at line 21 of file diag_matrix.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::diag_matrix ( const var_value< T > &  v )

inline

Return a square diagonal matrix with the specified vector of coefficients as the diagonal values.

Parameters

[in]

v

Specified vector.

Returns

Diagonal matrix with vector as diagonal values.

Definition at line 21 of file diag_matrix.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::diag_matrix ( T_x &&  x )

inline

Return a square diagonal matrix with the specified vector of coefficients as the diagonal values.

Template Parameters

T_x	type of input kernel generator expression for the diagonal

Parameters

x	input kernel generator expression for the diagonal

Returns

a kernel generator expression

Definition at line 25 of file diag_matrix.hpp.

template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_eigen_vector_t< T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>

auto stan::math::diag_post_multiply	(	const T1 &	m1,
		const T2 &	m2
	)

Return the product of the matrix and the diagonal matrix formed from the vector or row_vector.

Return the product of the matrix and a diagonal matrix formed from the vector or row_vector.

Template Parameters

T1	type of the matrix
T2	type of the vector/row_vector

Parameters

m1	input matrix
m2	input vector/row_vector

Returns

product of the matrix and the diagonal matrix formed from the vector or row_vector.

Definition at line 25 of file diag_post_multiply.hpp.

template<typename T1 , typename T2 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr>

auto stan::math::diag_post_multiply ( const T1 &  m1, const T2 &  m2  )

inline

Return the product of a matrix and the diagonal matrix formed from the vector or row_vector.

Template Parameters

T1	type of the matrix
T2	type of the vector/row_vector

Parameters

m1	input matrix
m2	input vector/row_vector

Returns

product of a matrix and the diagonal matrix formed from the vector or row_vector.

Definition at line 25 of file diag_post_multiply.hpp.

template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::diag_post_multiply ( T1 &&  v1, T2 &&  v2  )

inline

Return the product of a matrix and the diagonal matrix formed from the vector or row_vector.

Template Parameters

T1	type of the matrix
T2	type of the vector/row_vector

Parameters

v1	input matrix
v2	input vector/row_vector

Returns

product of a matrix and the diagonal matrix formed from the vector or row_vector.

Definition at line 30 of file diag_post_multiply.hpp.

template<typename T1 , typename T2 , require_eigen_vector_t< T1 > * = nullptr, require_eigen_t< T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>

auto stan::math::diag_pre_multiply	(	const T1 &	m1,
		const T2 &	m2
	)

Return the product of the diagonal matrix formed from the vector or row_vector and a matrix.

Template Parameters

T1	type of the vector/row_vector
T2	type of the matrix

Parameters

m1	input vector/row_vector
m2	input matrix

Returns

product of the diagonal matrix formed from the vector or row_vector and a matrix.

Definition at line 25 of file diag_pre_multiply.hpp.

template<typename T1 , typename T2 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr>

auto stan::math::diag_pre_multiply ( const T1 &  m1, const T2 &  m2  )

inline

Return the product of the diagonal matrix formed from the vector or row_vector and a matrix.

Template Parameters

T1	type of the vector/row_vector
T2	type of the matrix

Parameters

m1	input vector/row_vector
m2	input matrix

Returns

product of the diagonal matrix formed from the vector or row_vector and a matrix.

Definition at line 25 of file diag_pre_multiply.hpp.

template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::diag_pre_multiply ( T1 &&  v1, T2 &&  v2  )

inline

Return the product of the diagonal matrix formed from the vector or row_vector and a matrix.

Template Parameters

T1	type of the vector/row_vector
T2	type of the matrix

Parameters

v1	input vector/row_vector
v2	input matrix

Returns

product of the diagonal matrix formed from the vector or row_vector and a matrix.

Definition at line 30 of file diag_pre_multiply.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::diagonal ( const T &  m )

inline

Return a column vector of the diagonal elements of the specified matrix.

The matrix is not required to be square.

Template Parameters

T	type of the matrix

Parameters

m	Specified matrix.

Returns

Diagonal of the matrix.

Definition at line 19 of file diagonal.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::diagonal ( const var_value< T > &  M )

inline

Return a column vector of the diagonal elements of the specified matrix.

The matrix is not required to be square.

Parameters

M	Specified matrix.

Returns

Diagonal of the matrix.

Definition at line 21 of file diagonal.hpp.

var_value<matrix_cl<double> > stan::math::digamma ( const var_value< matrix_cl< double >> &  A )

inline

Returns the elementwise digamma() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise digamma() of the input.

Definition at line 18 of file digamma.hpp.

var stan::math::digamma ( const var &  a )

inline

Return the derivative of the log gamma function at the specified value.

Parameters

[in]

a

argument

Returns

derivative of log gamma function at argument

Definition at line 19 of file digamma.hpp.

template<typename T >

fvar<T> stan::math::digamma ( const fvar< T > &  x )

inline

Return the derivative of the log gamma function at the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

[in]

x

argument

Returns

derivative of the log gamma function at the specified argument

Definition at line 23 of file digamma.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::digamma ( const T &  a )

inline

Return the elementwise derivative of the log gamma function at the given input vector.

Template Parameters

T	a var_value with inner Eigen type

Parameters

[in]

a

vector

Returns

elementwise derivative of log gamma function

Definition at line 34 of file digamma.hpp.

double stan::math::digamma ( double  x )

inline

Return the derivative of the log gamma function at the specified value.

\[ \mbox{digamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{digamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)} \]

\[ \frac{\partial \, \Psi(x)}{\partial x} = \frac{\Gamma''(x)\Gamma(x)-(\Gamma'(x))^2}{\Gamma^2(x)} \]

The design follows the standard C++ library in returning NaN rather than throwing exceptions.

Parameters

[in]

x

argument

Returns

derivative of log gamma function at argument

Definition at line 47 of file digamma.hpp.

template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::digamma ( const T &  x )

inline

Vectorized version of digamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Digamma function applied to each value in x.

Exceptions

std::domain_error

if any value is a negative integer or 0

Definition at line 77 of file digamma.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

void stan::math::dims ( const T &  x, std::vector< int > &  result  )

inline

Pushes dimensions of given argument into given result vector.

For a scalar that is a no-op.

Template Parameters

type

of scalar

Parameters

x	argument
result	result

For an Eigen type those are the numbers of rows and columns.

Parameters

x	argument
result	result

Definition at line 20 of file dims.hpp.

template<typename T >

void stan::math::dims ( const var_value< T > &  x, std::vector< int > &  result  )

inline

Pushes dimensions of given argument into given result vector.

For var_value that is the dimensions of inner vari_value.

Template Parameters

type	in var_value

Parameters

x	argument
result	result

Definition at line 21 of file dims.hpp.

template<typename T >

void stan::math::dims ( const vari_value< T > &  x, std::vector< int > &  result  )

inline

Pushes dimensions of given argument into given result vector.

For vari_value containing Eigen type those are the numbers of rows and columns.

Parameters

x	argument
result	result

Definition at line 34 of file dims.hpp.

template<typename T , typename Alloc >

void stan::math::dims ( const std::vector< T, Alloc > &  x, std::vector< int > &  result  )

inline

Pushes dimensions of given argument into given result vector.

For a std::vector that is its size and dimensions of its elements.

Template Parameters

type	of scalar
Alloc	type of allocator

Parameters

x	argument
result	result

Definition at line 45 of file dims.hpp.

template<typename T >

std::vector<int> stan::math::dims ( const T &  x )

inline

Determines dimensions of an argument.

Parameters

x

argument

Returns

vector of sizes in each of arguemnt's dimensions

Definition at line 58 of file dims.hpp.

template<typename T_prob , typename T_prior_size >

return_type_t<T_prob, T_prior_size> stan::math::dirichlet_lpdf	(	const T_prob &	theta,
		const T_prior_size &	alpha
	)

Definition at line 130 of file dirichlet_lpdf.hpp.

template<typename T_prob , typename T_prior_size >

return_type_t<T_prob, T_prior_size> stan::math::dirichlet_lpmf	(	const T_prob &	theta,
		const T_prior_size &	alpha
	)

Definition at line 20 of file dirichlet_lpmf.hpp.

template<typename T_y , typename T_lower , typename T_upper >

double stan::math::discrete_range_lpmf ( const T_y &  y, const T_lower &  lower, const T_upper &  upper  )

inline

Definition at line 93 of file discrete_range_lpmf.hpp.

template<typename T1 , typename T2 , require_all_stan_scalar_t< T1, T2 > * = nullptr>

return_type_t< T1, T2 > stan::math::distance ( const T1 &  x1, const T2 &  x2  )

inline

Returns the distance between two scalars.

Returns the distance between the specified vectors.

Template Parameters

T1	type of first scalar.
T2	type of second scalar

Parameters

x1	First scalar.
x2	Second scalar.

Returns

Distance between two scalars

Exceptions

std::domain_error

If the arguments are not finite.

Template Parameters

T1	type of the first vector (must be derived from Eigen::MatrixBase and have one compile time dimension equal to 1)
T2	type of the second vector (must be derived from Eigen::MatrixBase and have one compile time dimension equal to 1)

Parameters

x1	First vector.
x2	Second vector.

Returns

Distance between the vectors.

Exceptions

std::domain_error

If the vectors are not the same size.

Definition at line 27 of file distance.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr>

auto stan::math::distance ( const T_a &  a, const T_b &  b  )

inline

Returns the distance between the specified vectors.

Template Parameters

T_a	type of the first kernel generator expression
T_b	type of the second kernel generator expression

Parameters

a	first kernel generator expression
b	second kernel generator expression

Returns

Distance between the inputs.

Exceptions

std::domain_error

If the matrices are not the same size

Definition at line 33 of file distance.hpp.

template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr>

return_type_t<Scal1, Scal2> stan::math::divide ( const Scal1 &  x, const Scal2 &  y  )

inline

Return the division of the first scalar by the second scalar.

Parameters

[in]	x	Specified scalar.
[in]	y	Specified scalar.

Returns

Scalar divided by the scalar.

Definition at line 24 of file divide.hpp.

int stan::math::divide ( int  x, int  y  )

inline

Definition at line 28 of file divide.hpp.

template<typename T_a , typename T_b , require_stan_scalar_t< T_b > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::divide ( T_a &&  a, T_b &&  b  )

inline

Elementwise division of a kernel generator expression and a scalar.

Template Parameters

T_a	type of the kernel generator expression
T_b	type of the scalar

Parameters

a	input kernel generator expression
b	scalar

Returns

Elementwise division of the input kernel generator expression with a scalar

Definition at line 32 of file divide.hpp.

template<typename T1 , typename T2 , require_any_eigen_t< T1, T2 > * = nullptr, require_all_not_st_var< T1, T2 > * = nullptr>

auto stan::math::divide ( const T1 &  m, const T2 &  c  )

inline

Return matrix divided by scalar.

Template Parameters

Mat	type of the matrix or expression
Scal	type of the scalar

Parameters

[in]	m	specified matrix or expression
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 46 of file divide.hpp.

template<typename Scalar , typename Mat , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr, require_all_st_var_or_arithmetic< Scalar, Mat > * = nullptr, require_any_st_var< Scalar, Mat > * = nullptr>

auto stan::math::divide ( const Mat &  m, Scalar  c  )

inline

Return matrix divided by scalar.

Template Parameters

Mat	A type inheriting from EigenBase with an Arithmetic scalar type.

Parameters

[in]	m	specified matrix or expression
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 126 of file operator_division.hpp.

template<typename Scalar , typename Mat , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scalar > * = nullptr, require_all_st_var_or_arithmetic< Scalar, Mat > * = nullptr, require_any_st_var< Scalar, Mat > * = nullptr>

auto stan::math::divide ( Scalar  c, const Mat &  m  )

inline

Return scalar divided by matrix.

Template Parameters

Mat	Either a type inheriting from EigenBase with a scalar type of var or a var_value<T> with type T inheriting from EigenBase.

Parameters

[in]	m	specified matrix or expression
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 170 of file operator_division.hpp.

template<typename Mat1 , typename Mat2 , require_all_matrix_st< is_var_or_arithmetic, Mat1, Mat2 > * = nullptr, require_any_matrix_st< is_var, Mat1, Mat2 > * = nullptr>

auto stan::math::divide ( const Mat1 &  m1, const Mat2 &  m2  )

inline

Return a matrix divided by a matrix elementwise.

Template Parameters

Mat1	Either a type inheriting from EigenBase or a var_value<T> with type T inheriting from EigenBase.
Mat2	Either a type inheriting from EigenBase or a var_value<T> with type T inheriting from EigenBase.

Parameters

[in]	m1	specified matrix or expression
[in]	m2	specified matrix or expression

Definition at line 215 of file operator_division.hpp.

template<typename T_x , typename T_v >

std::vector< Eigen::Matrix<return_type_t<T_x, T_v, double>, Eigen::Dynamic, 1> > stan::math::divide_columns ( const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &  x, const std::vector< T_v > &  vec  )

inline

Takes Stan data type vector[n] x[D] and divides column vector in x element-wise by the values in vec.

Template Parameters

T_x	Type of dividend
T_v	Scalar type of divisor

Parameters

x	std::vector of matrices
vec	std::vector of divisors

Exceptions

std::invalid

argument if D != length of vector

Definition at line 25 of file divide_columns.hpp.

template<typename T_shape >

return_type_t<double, T_shape> stan::math::do_lkj_constant	(	const T_shape &	eta,
		const unsigned int &	K
	)

Definition at line 16 of file lkj_corr_lpdf.hpp.

template<typename T >

void stan::math::domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Deprecated:

use throw_domain_error

Definition at line 14 of file domain_error.hpp.

template<typename T >

void stan::math::domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Deprecated:

use throw_domain_error

Definition at line 23 of file domain_error.hpp.

template<typename T >

void stan::math::domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Deprecated:

use throw_domain_error_vec

Definition at line 13 of file domain_error_vec.hpp.

template<typename T >

void stan::math::domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1  )

inline

Deprecated:

use throw_domain_error_vec

Definition at line 22 of file domain_error_vec.hpp.

double stan::math::dot ( const std::vector< double > &  x, const std::vector< double > &  y  )

inline

Definition at line 11 of file dot.hpp.

template<typename Vec1 , typename Vec2 , require_all_eigen_vector_t< Vec1, Vec2 > * = nullptr, require_not_var_t< return_type_t< Vec1, Vec2 >> * = nullptr>

auto stan::math::dot_product ( const Vec1 &  v1, const Vec2 &  v2  )

inline

Returns the dot product of the specified vectors.

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::invalid_argument

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 24 of file dot_product.hpp.

template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::dot_product ( const T_a &  a, const T_b &  b  )

inline

Returns the dot product of the specified vectors.

Template Parameters

T_a	type of the first vector
T_b	type of the second vector

Parameters

a	First vector.
b	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::invalid_argument

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 26 of file dot_product.hpp.

template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

var stan::math::dot_product ( T1 &&  v1, T2 &&  v2  )

inline

Returns the dot product.

Template Parameters

T1	type of the first vector
T2	type of the second vector

Parameters

[in]	v1	First vector.
[in]	v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

if sizes of v1 and v2 do not match.

Definition at line 28 of file dot_product.hpp.

template<typename T1 , typename T2 , require_all_vector_t< T1, T2 > * = nullptr, require_not_complex_t< return_type_t< T1, T2 >> * = nullptr, require_all_not_std_vector_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>

var stan::math::dot_product ( const T1 &  v1, const T2 &  v2  )

inline

Returns the dot product.

Template Parameters

T1	type of elements in the first vector
T2	type of elements in the second vector

Parameters

[in]	v1	First vector.
[in]	v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

if sizes of v1 and v2 do not match.

Definition at line 38 of file dot_product.hpp.

template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr, require_all_not_var_t< Scalar1, Scalar2 > * = nullptr>

auto stan::math::dot_product ( const Scalar1 *  v1, const Scalar2 *  v2, size_t  length  )

inline

Returns the dot product of the specified arrays.

Parameters

v1	First array.
v2	Second array.
length	Length of both arrays.

Definition at line 39 of file dot_product.hpp.

template<typename Scalar1 , typename Scalar2 , typename Alloc1 , typename Alloc2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>

return_type_t<Scalar1, Scalar2> stan::math::dot_product ( const std::vector< Scalar1, Alloc1 > &  v1, const std::vector< Scalar2, Alloc2 > &  v2  )

inline

Returns the dot product of the specified arrays.

Parameters

v1	First array.
v2	Second array.

Exceptions

std::domain_error

if the vectors are not the same size.

Definition at line 56 of file dot_product.hpp.

double stan::math::dot_self ( const std::vector< double > &  x )

inline

Definition at line 13 of file dot_self.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::dot_self ( const var_value< T > &  v )

inline

Returns the dot product of a vector of var with itself.

Template Parameters

T	type of the vector

Parameters

[in]

v

Vector.

Returns

Dot product of the vector with itself.

Definition at line 21 of file dot_self.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

auto stan::math::dot_self ( const T &  a )

inline

Returns squared norm of a vector or matrix.

For vectors that equals the dot product of the specified vector with itself.

Template Parameters

T	type of the vector

Parameters

a

Vector.

Definition at line 21 of file dot_self.hpp.

template<typename T , require_eigen_vector_vt< is_var, T > * = nullptr>

var stan::math::dot_self ( const T &  v )

inline

Returns the dot product of a vector of var with itself.

Returns the dot product of a var_value<Vector> with itself.

Template Parameters

T	type of the vector (must have one compile time dimension equal to 1)

Parameters

[in]

v

Vector.

Returns

Dot product of the vector with itself.

Template Parameters

A	var_value<> whose inner type has one compile time row or column.

Parameters

[in]

v

Vector.

Returns

Dot product of the vector with itself.

Definition at line 24 of file dot_self.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_var, T > * = nullptr>

value_type_t<T> stan::math::dot_self ( const T &  v )

inline

Returns squared norm of a vector or matrix.

For vectors that equals the dot product of the specified vector with itself.

Template Parameters

T	type of the vector (must be derived from Eigen::MatrixBase)

Parameters

v

Vector.

Definition at line 30 of file dot_self.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::double_exponential_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 110 of file double_exponential_lpdf.hpp.

static constexpr double stan::math::e ( )

static

Return the base of the natural logarithm.

Returns

Base of natural logarithm.

Definition at line 20 of file constants.hpp.

static constexpr double stan::math::egamma ( )

static

Return the Euler's gamma constant.

Returns

Euler's Gamma.

Definition at line 27 of file constants.hpp.

template<typename T >

Eigen::Matrix<std::complex<T>, -1, 1> stan::math::eigenvalues

(

const Eigen::Matrix< T,-1,-1 > &

m

)

Definition at line 11 of file eigenvalues.hpp.

matrix_cl<double> stan::math::eigenvalues_sym ( const matrix_cl< double > &  m )

inline

Definition at line 11 of file eigenvalues_sym.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, 1> stan::math::eigenvalues_sym

(

const EigMat &

m

)

Return the eigenvalues of the specified symmetric matrix in descending order of magnitude.

This function is more efficient than the general eigenvalues function for symmetric matrices.

See eigen_decompose() for more information.

Template Parameters

EigMat

type of the matrix

Parameters

m	Specified matrix.

Returns

Eigenvalues of matrix.

Definition at line 24 of file eigenvalues_sym.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::eigenvalues_sym ( const T &  m )

inline

Return the eigenvalues of the specified symmetric matrix.

See eigen_decompose() for more information.

Template Parameters

T	type of input matrix.

Parameters

m	Specified matrix.

Returns

Eigenvalues of matrix.

Definition at line 25 of file eigenvalues_sym.hpp.

template<typename T >

Eigen::Matrix<std::complex<T>, -1, -1> stan::math::eigenvectors

(

const Eigen::Matrix< T,-1,-1 > &

m

)

Definition at line 11 of file eigenvectors.hpp.

matrix_cl<double> stan::math::eigenvectors_sym ( const matrix_cl< double > &  m )

inline

Definition at line 11 of file eigenvectors_sym.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::eigenvectors_sym

(

const EigMat &

m

)

Definition at line 14 of file eigenvectors_sym.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::eigenvectors_sym ( const T &  m )

inline

Return the eigenvectors of the specified symmetric matrix.

See eigen_decompose() for more information.

Template Parameters

T	type of input matrix.

Parameters

m	Specified matrix.

Returns

Eigenvectors of matrix.

Definition at line 25 of file eigenvectors_sym.hpp.

template<typename F , typename T , typename... Indexings, require_stan_scalar_t< T > * = nullptr>

void stan::math::elementwise_check ( const F &  is_good, const char *  function, const char *  name, const T &  x, const char *  must_be, const Indexings &...  indexings  )

inline

Check that the predicate holds for the value of x.

Check that the predicate holds for all elements of the value of x.

This overload works on scalars.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

This overload works on std::vector types.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

This overload works on vars containing Eigen types.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

Definition at line 116 of file elementwise_check.hpp.

template<typename F , typename T , typename... Indexings, require_eigen_t< T > * = nullptr, std::enable_if_t<(Eigen::internal::traits< T >::Flags &Eigen::LinearAccessBit)||T::IsVectorAtCompileTime > * = nullptr>

void stan::math::elementwise_check ( const F &  is_good, const char *  function, const char *  name, const T &  x, const char *  must_be, const Indexings &...  indexings  )

inline

Check that the predicate holds for all elements of the value of x.

Check that the predicate holds for all the elements of the value of x.

This overload works on Eigen types that support linear indexing.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

This overload works on col-major Eigen types that do not support linear indexing.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

This overload works on row-major Eigen types that do not support linear indexing.

Template Parameters

F	type of predicate
T	type of x
Indexings	types of indexings

Parameters

is_good	predicate to check, must accept doubles and produce bools
function	function name (for error messages)
name	variable name (for error messages)
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc
must_be	message describing what the value should be
indexings	any additional indexing to print. Intended for internal use in elementwise_check only.

Exceptions

<tt>std::domain_error</tt>

if is_good returns false for the value of any element in x

Definition at line 150 of file elementwise_check.hpp.

template<typename F , typename T >

bool stan::math::elementwise_is ( const F &  is_good, const T &  x  )

inline

Check that the predicate holds for the value of x, working elementwise on containers.

If x is a scalar, check the double underlying the scalar. If x is a container, check each element inside x, recursively.

Template Parameters

F	type of predicate
T	type of x

Parameters

is_good	predicate to check, must accept doubles and produce bools
x	variable to check, can be a scalar, a container of scalars, a container of containers of scalars, etc

Returns

false if any of the scalars fail the error check

Definition at line 333 of file elementwise_check.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>

auto stan::math::elt_divide	(	const Mat1 &	m1,
		const Mat2 &	m2
	)

Return the elementwise division of the specified matrices.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	First matrix or expression
m2	Second matrix or expression

Returns

Elementwise division of matrices.

Template Parameters

Mat1	type of the first matrix
Mat2	type of the second matrix

Parameters

m1	First matrix
m2	Second matrix

Returns

Elementwise division of matrices.

Definition at line 25 of file elt_divide.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::elt_divide ( T_a &&  a, T_b &&  b  )

inline

Elementwise division of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Elementwise division of the input arguments

Definition at line 31 of file elt_divide.hpp.

template<typename Mat , typename Scal , require_matrix_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr>

auto stan::math::elt_divide	(	const Mat &	m,
		Scal	s
	)

Return the elementwise division of the specified matrix by the specified scalar.

Template Parameters

Mat	type of the matrix or expression
Scal	type of the scalar

Parameters

m	matrix or expression
s	scalar

Returns

Elementwise division of a scalar by matrix.

Definition at line 43 of file elt_divide.hpp.

template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr>

auto stan::math::elt_divide	(	Scal	s,
		const Mat &	m
	)

Return the elementwise division of the specified scalar by the specified matrix.

Template Parameters

Scal	type of the scalar
Mat	type of the matrix or expression

Parameters

s	scalar
m	matrix or expression

Returns

Elementwise division of a scalar by matrix.

Template Parameters

Scal	type of the scalar
Mat	type of the matrix

Parameters

s	scalar
m	matrix or expression

Returns

Elementwise division of a scalar by matrix.

Definition at line 60 of file elt_divide.hpp.

template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr>

auto stan::math::elt_divide	(	Scal1	s1,
		Scal2	s2
	)

Definition at line 66 of file elt_divide.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>

auto stan::math::elt_multiply	(	const Mat1 &	m1,
		const Mat2 &	m2
	)

Return the elementwise multiplication of the specified matrices.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	First matrix or expression
m2	Second matrix or expression

Returns

Elementwise product of matrices.

Definition at line 26 of file elt_multiply.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::elt_multiply ( T_a &&  a, T_b &&  b  )

inline

Elementwise multiplication of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Elementwise multiplication of the input arguments

Definition at line 31 of file elt_multiply.hpp.

template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>

auto stan::math::elt_multiply	(	const Scalar1 &	a,
		const Scalar2 &	b
	)

Return the elementwise multiplication of the specified scalars.

Template Parameters

Mat1	type of the first scalar
Mat2	type of the second scalar

Parameters

a	First scalar
b	Second scalar

Returns

Product of scalars

Definition at line 44 of file elt_multiply.hpp.

template<typename T1 , typename T2 , require_any_matrix_t< T1, T2 > * = nullptr, require_any_stan_scalar_t< T1, T2 > * = nullptr>

auto stan::math::elt_multiply ( const T1 &  A, const T2 &  B  )

inline

Return specified matrix multiplied by specified scalar.

One argument is a matrix and one is a scalar.

Template Parameters

T1	type of the first argument
T2	type of the second argument

Parameters

A	first argument
B	second argument

Returns

product of matrix and scalar

Definition at line 62 of file elt_multiply.hpp.

static bool stan::math::empty_nested ( )

inlinestatic

Return true if there is no nested autodiff being executed.

Definition at line 12 of file empty_nested.hpp.

template<typename T >

fvar<T> stan::math::erf ( const fvar< T > &  x )

inline

Definition at line 15 of file erf.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::erf ( const var_value< T > &  A )

inline

Returns the elementwise erf() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise erf() of the input.

Definition at line 20 of file erf.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::erf ( const T &  x )

inline

Returns the elementwise erf() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

T	type of container

Parameters

x

container

Returns

Error function applied to each value in x.

Definition at line 38 of file erf.hpp.

var stan::math::erf ( const var &  a )

inline

The error function for variables (C99).

The derivative is

\(\frac{d}{dx} \mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \exp(-x^2)\).

\[ \mbox{erf}(x) = \begin{cases} \operatorname{erf}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erf}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erf}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erf}(x)}{\partial x} = \frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters

a	The variable.

Returns

Error function applied to the variable.

Definition at line 48 of file erf.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::erf ( const var_value< T > &  a )

inline

Definition at line 56 of file erf.hpp.

template<typename T >

fvar<T> stan::math::erfc ( const fvar< T > &  x )

inline

Definition at line 15 of file erfc.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::erfc ( const var_value< T > &  A )

inline

Returns the elementwise erfc() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise erfc() of the input.

Definition at line 20 of file erfc.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::erfc ( const T &  x )

inline

Returns the elementwise erfc() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

T	type of container

Parameters

x

container

Returns

Complementary error function applied to each value in x.

Definition at line 39 of file erfc.hpp.

var stan::math::erfc ( const var &  a )

inline

The complementary error function for variables (C99).

The derivative is

\(\frac{d}{dx} \mbox{erfc}(x) = - \frac{2}{\sqrt{\pi}} \exp(-x^2)\).

\[ \mbox{erfc}(x) = \begin{cases} \operatorname{erfc}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erfc}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erfc}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erfc}(x)}{\partial x} = -\frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters

a	The variable.

Returns

Complementary error function applied to the variable.

Definition at line 48 of file erfc.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::erfc ( const var_value< T > &  a )

inline

Definition at line 56 of file erfc.hpp.

template<typename T , require_same_t< std::decay_t< T >, plain_type_t< T >> * = nullptr>

T stan::math::eval ( T &&  arg )

inline

Inputs which have a plain_type equal to the own time are forwarded unmodified (for Eigen expressions these types are different)

Template Parameters

T	Input type

Parameters

[in]

arg

Input argument

Returns

Forwarded input argument

Definition at line 20 of file eval.hpp.

template<typename T , require_not_same_t< std::decay_t< T >, plain_type_t< T >> * = nullptr>

decltype(auto) stan::math::eval ( const T &  arg )

inline

Inputs which have a plain_type different from their own type are Eval'd (this catches Eigen expressions)

Template Parameters

T	Input type

Parameters

[in]

arg

Input argument

Returns

Eval'd argument

Definition at line 34 of file eval.hpp.

template<typename T >

fvar<T> stan::math::exp ( const fvar< T > &  x )

inline

Definition at line 13 of file exp.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::exp ( const var_value< T > &  A )

inline

Returns the elementwise exp() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise exp() of the input.

Definition at line 20 of file exp.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::exp ( const std::complex< fvar< T >> &  z )

inline

Return the natural exponentiation (base e) of the specified complex number.

Template Parameters

T	value type of autodiff variable

Parameters

z	complex argument

Returns

exponentiation of argument

Definition at line 26 of file exp.hpp.

var stan::math::exp ( const var &  a )

inline

Return the exponentiation of the specified variable (cmath).

\[ \mbox{exp}(x) = \begin{cases} e^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable to exponentiate.

Returns

Exponentiated variable.

Definition at line 42 of file exp.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>

auto stan::math::exp ( const Container &  x )

inline

Return the elementwise exp() of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

Container

type of container

Parameters

[in]

x

container

Returns

Elementwise application of exponentiation to the argument.

Definition at line 47 of file exp.hpp.

std::complex<var> stan::math::exp ( const std::complex< var > &  z )

inline

Return the exponentiation (base e) of the specified complex number.

Parameters

z

argument

Returns

exponentiation of argument

Definition at line 53 of file exp.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::exp ( const Container &  x )

inline

Version of exp() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Elementwise application of exponentiation to the argument.

Definition at line 61 of file exp.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::exp ( const T &  x )

inline

Return the exponentiation of the elements of x.

Template Parameters

T

type of x

Parameters

x

argument

Returns

elementwise exponentiation of x

Definition at line 65 of file exp.hpp.

template<typename T >

fvar<T> stan::math::exp2 ( const fvar< T > &  x )

inline

Definition at line 14 of file exp2.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::exp2 ( const var_value< T > &  A )

inline

Returns the elementwise exp2() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise exp2() of the input.

Definition at line 20 of file exp2.hpp.

var stan::math::exp2 ( const var &  a )

inline

Exponentiation base 2 function for variables (C99).

The derivative is

\(\frac{d}{dx} 2^x = (\log 2) 2^x\).

\[ \mbox{exp2}(x) = \begin{cases} 2^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp2}(x)}{\partial x} = \begin{cases} 2^x\ln2 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	The variable.

Returns

Two to the power of the specified variable.

Definition at line 39 of file exp2.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::exp2 ( const T &  x )

inline

Return the elementwise exp2() of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise exp2 of members of container.

Definition at line 42 of file exp2.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

auto stan::math::exp2 ( const var_value< T > &  a )

inline

Definition at line 46 of file exp2.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 29 of file exp_mod_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 31 of file exp_mod_normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 31 of file exp_mod_normal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_inv_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 29 of file exp_mod_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma, const T_inv_scale &  lambda  )

inline

Definition at line 122 of file exp_mod_normal_lpdf.hpp.

template<typename T >

fvar<T> stan::math::expm1 ( const fvar< T > &  x )

inline

Definition at line 13 of file expm1.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::expm1 ( const var_value< T > &  A )

inline

Returns the elementwise expm1() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise expm1() of the input.

Definition at line 20 of file expm1.hpp.

var stan::math::expm1 ( const var &  a )

inline

The exponentiation of the specified variable minus 1 (C99).

The derivative is given by

\(\frac{d}{dx} \exp(a) - 1 = \exp(a)\).

\[ \mbox{expm1}(x) = \begin{cases} e^x-1 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{expm1}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	The variable.

Returns

Two to the power of the specified variable.

Definition at line 38 of file expm1.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::expm1 ( const T &  x )

inline

Return the elementwise expm1() of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural exponential of each value in x minus one.

Definition at line 39 of file expm1.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

auto stan::math::expm1 ( const var_value< T > &  a )

inline

Definition at line 45 of file expm1.hpp.

template<typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>

return_type_t<T_y, T_inv_scale> stan::math::exponential_lccdf	(	const T_y &	y,
		const T_inv_scale &	beta
	)

Definition at line 20 of file exponential_lccdf.hpp.

template<typename T_y , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_inv_scale > * = nullptr>

return_type_t<T_y, T_inv_scale> stan::math::exponential_lcdf	(	const T_y &	y,
		const T_inv_scale &	beta
	)

Definition at line 24 of file exponential_lcdf.hpp.

template<typename T_y , typename T_inv_scale >

return_type_t<T_y, T_inv_scale> stan::math::exponential_lpdf ( const T_y &  y, const T_inv_scale &  beta  )

inline

Definition at line 106 of file exponential_lpdf.hpp.

template<typename T >

T stan::math::F32	(	const T &	a1,
		const T &	a2,
		const T &	a3,
		const T &	b1,
		const T &	b2,
		const T &	z,
		double	precision = 1e-6,
		int	max_steps = 1e5
	)

Hypergeometric function (3F2).

Function reference: http://dlmf.nist.gov/16.2

\[ _3F_2 \left( \begin{matrix}a_1 a_2 a3 \\ b_1 b_2\end{matrix}; z \right) = \sum_k=0^\infty \frac{(a_1)_k(a_2)_k(a_3)_k}{(b_1)_k(b_2)_k}\frac{z^k}{k!} \]

Where _k$ is an upper shifted factorial.

Calculate the hypergeometric function (3F2) as the power series directly to within precision or until max_steps terms.

This function does not have a closed form but will converge if:

• |z| is less than 1
• |z| is equal to one and b1 + b2 < a1 + a2 + a3 This function is a rational polynomial if
• a1, a2, or a3 is a non-positive integer This function can be treated as a rational polynomial if
• b1 or b2 is a non-positive integer and the series is terminated prior to the final term.

Template Parameters

T	type of arguments and result

Parameters

[in]	a1	a1 (always called with 1 from beta binomial cdfs)
[in]	a2	a2 (always called with a2 > 1)
[in]	a3	a3 (always called with int a3 <= 0)
[in]	b1	b1 (always called with int b1 < |a3|)
[in]	b2	b2 (always <= 1)
[in]	z	z (is always called with 1 from beta binomial cdfs)
[in]	precision	precision of the infinite sum. defaults to 1e-6
[in]	max_steps	number of steps to take. defaults to 1e5

Definition at line 53 of file F32.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

auto stan::math::fabs

(

T

x

)

Definition at line 14 of file fabs.hpp.

template<typename T >

fvar<T> stan::math::fabs ( const fvar< T > &  x )

inline

Definition at line 15 of file fabs.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::fabs ( const var_value< T > &  A )

inline

Returns the elementwise fabs() of the input var_value<matrix_cl<double>>.

Parameters

A	input var_value<matrix_cl<double>>.

Returns

Elementwise fabs() of the input argument.

Definition at line 21 of file fabs.hpp.

var stan::math::fabs ( const var &  a )

inline

Return the absolute value of the variable (cmath).

Choosing an arbitrary value at the non-differentiable point 0,

\(\frac{d}{dx}|x| = \mbox{sgn}(x)\).

where \(\mbox{sgn}(x)\) is the signum function, taking values -1 if \(x < 0\), 0 if \(x == 0\), and 1 if \(x == 1\).

The function abs() provides the same behavior, with abs() defined in stdlib.h and fabs() defined in cmath. The derivative is 0 if the input is 0.

Returns std::numeric_limits<double>::quiet_NaN() for NaN inputs.

\[ \mbox{fabs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fabs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Absolute value of variable.

Definition at line 50 of file fabs.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_stan_scalar_t< Container > * = nullptr>

auto stan::math::fabs ( const Container &  x )

inline

Returns the elementwise fabs() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

Absolute value of each value in x.

Definition at line 51 of file fabs.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::fabs ( const Container &  x )

inline

Version of fabs() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Absolute value of each value in x.

Definition at line 65 of file fabs.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::fabs ( const VarMat &  a )

inline

Return the absolute value of the variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Input variable.

Returns

Absolute value of variable.

Definition at line 71 of file fabs.hpp.

template<typename T_Sigma , typename T_CPCs , typename T_sds , require_eigen_t< T_Sigma > * = nullptr, require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_all_vt_same< T_Sigma, T_CPCs, T_sds > * = nullptr>

bool stan::math::factor_cov_matrix	(	const T_Sigma &	Sigma,
		T_CPCs &&	CPCs,
		T_sds &&	sds
	)

This function is intended to make starting values, given a covariance matrix Sigma.

The transformations are hard coded as log for standard deviations and Fisher transformations (atanh()) of CPCs

Template Parameters

T	type of elements in the matrix and arrays

Parameters

[in]	Sigma	covariance matrix
[out]	CPCs	fill this unbounded (does not resize)
[out]	sds	fill this unbounded

Returns

false if any of the diagonals of Sigma are 0

Definition at line 28 of file factor_cov_matrix.hpp.

template<typename T_U , typename T_CPCs , require_eigen_t< T_U > * = nullptr, require_eigen_vector_t< T_CPCs > * = nullptr, require_vt_same< T_U, T_CPCs > * = nullptr>

void stan::math::factor_U	(	const T_U &	U,
		T_CPCs &&	CPCs
	)

This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix.

Template Parameters

T	type of elements in the matrix

Parameters

U	Sigma matrix
CPCs	fill this unbounded

Definition at line 25 of file factor_U.hpp.

var stan::math::falling_factorial ( const var &  a, int  b  )

inline

Definition at line 12 of file falling_factorial.hpp.

template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_st_integral< T2 > * = nullptr>

auto stan::math::falling_factorial ( const var_value< T1 > &  a, const T2 &  b  )

inline

Definition at line 22 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::falling_factorial ( const fvar< T > &  x, int  n  )

inline

Return autodiff variable with the gradient and result of the falling factorial function applied to the inputs.

Will throw for NaN x and for negative n, as implemented in primitive function.

Template Parameters

T	inner type of the fvar

Parameters

x	Argument.
n	Argument

Returns

tangent of falling factorial at arguments.

Definition at line 26 of file falling_factorial.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

return_type_t<T> stan::math::falling_factorial ( const T &  x, int  n  )

inline

Return the falling factorial function evaluated at the inputs.

Will throw for NaN x and for negative n

Template Parameters

T	Type of x argument.

Parameters

x	Argument.
n	Argument

Returns

Result of falling factorial function.

Exceptions

std::domain_error	if x is NaN
std::domain_error	if n is negative

\[ \mbox{falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ (x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)} \]

\[ \frac{\partial \, (x)_n}{\partial x} = (x)_n\Psi(x+1) \]

\[ \frac{\partial \, (x)_n}{\partial n} = -(x)_n\Psi(n+1) \]

Definition at line 64 of file falling_factorial.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::falling_factorial ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the falling factorial function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Falling factorial function applied to the two inputs.

Definition at line 83 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::fdim ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	inner type of the fvar

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 21 of file fdim.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

double stan::math::fdim ( T1  x, T2  y  )

inline

Return the positive difference of the specified values (C++11).

The function is defined by

fdim(x, y) = (x > y) ? (x - y) : 0.

Parameters

x	First value.
y	Second value.

Returns

max(x- y, 0)

Definition at line 22 of file fdim.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::fdim ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the fdim function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Fdim function applied to the two inputs.

Definition at line 38 of file fdim.hpp.

template<typename T >

fvar<T> stan::math::fdim ( const fvar< T > &  x, double  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	inner type of the fvar

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 39 of file fdim.hpp.

template<typename T >

fvar<T> stan::math::fdim ( double  x, const fvar< T > &  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	inner type of the fvar

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 57 of file fdim.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::fdim ( T_a &&  a, T_b &&  b  )

inline

Return the positive difference between the first variable's the value and the second's (C99, C++11).

The function values and derivatives are defined by

\[ \mbox{fdim}(x, y) = \begin{cases} x-y & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

The positive difference between the first and second variable.

Definition at line 61 of file fdim.hpp.

var stan::math::fdim ( const var &  a, const var &  b  )

inline

Return the positive difference between the first variable's the value and the second's (C99, C++11).

The function values and derivatives are defined by

\[ \mbox{fdim}(x, y) = \begin{cases} x-y & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

The positive difference between the first and second variable.

Definition at line 91 of file fdim.hpp.

var stan::math::fdim ( double  a, const var &  b  )

inline

Return the positive difference between the first value and the value of the second variable (C99, C++11).

See fdim(var, var) for definitions of values and derivatives.

Parameters

a	First value.
b	Second variable.

Returns

The positive difference between the first and second arguments.

Definition at line 110 of file fdim.hpp.

var stan::math::fdim ( const var &  a, double  b  )

inline

Return the positive difference between the first variable's value and the second value (C99, C++11).

See fdim(var, var) for definitions of values and derivatives.

Parameters

a	First value.
b	Second variable.

Returns

The positive difference between the first and second arguments.

Definition at line 127 of file fdim.hpp.

template<typename EigMat , typename S , require_eigen_t< EigMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>

void stan::math::fill ( EigMat &  x, const S &  y  )

inline

Fill the specified container with the specified value.

The specified matrix is filled by element.

Template Parameters

EigMat	Type inheriting from EigenBase
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 24 of file fill.hpp.

template<typename VarMat , typename S , require_var_matrix_t< VarMat > * = nullptr, require_var_t< S > * = nullptr>

void stan::math::fill ( VarMat &  x, const S &  y  )

inline

Fill the specified container with the specified value.

The specified matrix is filled by element.

Template Parameters

VarMat	a var_value with inner type from EigenBase
S	A var.

Parameters

x	Container.
y	Value.

The specified matrix is filled by element.

Template Parameters

VarMat	a var_value with inner type from EigenBase
S	An arithmetic type.

Parameters

x	Container.
y	Value.

Definition at line 25 of file fill.hpp.

template<typename T , typename S , require_t< std::is_assignable< std::decay_t< T > &, std::decay_t< S >>> * = nullptr>

void stan::math::fill ( T &  x, S &&  y  )

inline

Fill the specified container with the specified value.

This base case simply assigns the value to the container.

Template Parameters

T	Type of reference container.
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 41 of file fill.hpp.

template<typename Vec , typename S , require_std_vector_t< Vec > * = nullptr>

void stan::math::fill ( Vec &  x, S &&  y  )

inline

Fill the specified container with the specified value.

Each container in the specified standard vector is filled recursively by calling fill.

Template Parameters

Vec	A standard vector
S	type of value

Parameters

[in]	x	Container.
[in,out]	y	Value.

Definition at line 57 of file fill.hpp.

template<typename F >

void stan::math::finite_diff_grad_hessian	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::MatrixXd &	hess,
		std::vector< Eigen::MatrixXd > &	grad_hess_fx,
		double	epsilon = 1e-04
	)

Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, Eigen::Dynamic, 1>&)

Reference:

De Levie: An improved numerical approximation for the first derivative, page 3

4 calls to the function, f.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	hess	Hessian matrix
[out]	grad_hess_fx	gradient of Hessian of function at argument
[in]	epsilon	perturbation size

Definition at line 41 of file finite_diff_grad_hessian.hpp.

template<typename F >

void stan::math::finite_diff_grad_hessian_auto	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::MatrixXd &	hess,
		std::vector< Eigen::MatrixXd > &	grad_hess_fx
	)

Calculate the value, Hessian, and the gradient of the Hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference.

The functor must implement

double operator()(const Eigen::VectorXd&) const;

Reference for finite difference to compute gradient:

De Levie: An improved numerical approximation for the first derivative, page 3

Step size for dimension i is set automatically using stan::math::finite_diff_stepsize(H(i, j)); the nested finite differences are over entries in the Hessian.

Evaluating this function involves 6 calls to the Hessian autodiff function for each entry in the Hessian.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	hess	Hessian matrix
[out]	grad_hess_fx	gradient of Hessian of function at argument

Definition at line 44 of file finite_diff_grad_hessian_auto.hpp.

template<typename F >

void stan::math::finite_diff_gradient	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx,
		double	epsilon = 1e-03
	)

Calculate the value and the gradient of the specified function at the specified argument using finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, -1, 1>&) const;

If epsilon is chosen to be near the square root of the machine precision and the input vector elements are all roughly unit scale, and if the function has reasonable limits on variation, error should be on the order of epsilon^6.

The reference for the algorithm is:

Robert de Levie. 2009. An improved numerical approximation for the first derivative. Journal of Chemical Sciences 121(5), page 3.

Evaluating this function involves 6 calls to f for each dimension of the input.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument
[in]	epsilon	perturbation size

Definition at line 43 of file finite_diff_gradient.hpp.

template<typename F >

void stan::math::finite_diff_gradient_auto	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument using finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, -, 1>&) const;

Error of derivative in dimension i should be on the should be on order of epsilon(i)^6, where epsilon(i) = sqrt(delta) * abs(x(i)) for input x at dimension i.

The reference for this algorithm is:

Robert de Levie. 2009. An improved numerical approximation for the first derivative. Journal of Chemical Sciences 121(5), page 3.

The reference for automatically setting the difference is this section of the Wikipedia,

Numerical differentiation: practical considerations using floating point arithmetic.

Evaluating this function involves 6 calls to the function being differentiated for each dimension in the input, plus one global evaluation. All evaluations will be for double-precision inputs.

Template Parameters

F	Type of function

Parameters

[in]	f	function
[in]	x	argument to function
[out]	fx	function applied to argument
[out]	grad_fx	gradient of function at argument

Definition at line 50 of file finite_diff_gradient_auto.hpp.

double stan::math::finite_diff_stepsize ( double  u )

inline

Return the stepsize for finite difference evaluations at the specified scalar.

The formula used is stepsize(u) = cbrt(epsilon) * max(1, abs(u)).

Parameters

u	initial value to increment

Returns

stepsize away from u for finite differences

Definition at line 21 of file finite_diff_stepsize.hpp.

template<typename T >

fvar<T> stan::math::floor ( const fvar< T > &  x )

inline

Definition at line 12 of file floor.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::floor ( const var_value< T > &  A )

inline

Returns the elementwise floor() of the input var_value<matrix_cl<double>>.

Parameters

A	input var_value<matrix_cl<double>>.

Returns

Elementwise floor() of the input argument.

Definition at line 21 of file floor.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>

auto stan::math::floor ( const Container &  x )

inline

Returns the elementwise floor() of the input, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

Greatest integer <= each value in x.

Definition at line 41 of file floor.hpp.

var stan::math::floor ( const var &  a )

inline

Return the floor of the specified variable (cmath).

The derivative of the floor function is defined and zero everywhere but at integers, so we set these derivatives to zero for convenience,

\(\frac{d}{dx} {\lfloor x \rfloor} = 0\).

The floor function rounds down. For double values, this is the largest integral value that is not greater than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{floor}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{floor}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Floor of the variable.

Definition at line 47 of file floor.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

auto stan::math::floor ( const var_value< T > &  a )

inline

Definition at line 50 of file floor.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>

auto stan::math::floor ( const Container &  x )

inline

Version of floor() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Greatest integer <= each value in x.

Definition at line 56 of file floor.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_arithmetic_t< T1, T2, T3 > * = nullptr>

double stan::math::fma ( T1  x, T2  y, T3  z  )

inline

Return the product of the first two arguments plus the third argument.

Warning: This does not delegate to the high-precision platform-specific fma() implementation.

Parameters

x	First argument.
y	Second argument.
z	Third argument.

Returns

The product of the first two arguments plus the third argument.

Definition at line 26 of file fma.hpp.

var stan::math::fma ( const var &  x, const var &  y, const var &  z  )

inline

The fused multiply-add function for three variables (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + z = y\), and

\(\frac{\partial}{\partial y} (x * y) + z = x\), and

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 31 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_any_matrix_t< T1, T2, T3 > * = nullptr, require_not_var_t< return_type_t< T1, T2, T3 >> * = nullptr>

auto stan::math::fma ( T1 &&  x, T2 &&  y, T3 &&  z  )

inline

Definition at line 34 of file fma.hpp.

template<typename Tc , require_arithmetic_t< Tc > * = nullptr>

var stan::math::fma ( const var &  x, const var &  y, Tc &&  z  )

inline

The fused multiply-add function for two variables and a value (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + z = y\), and

\(\frac{\partial}{\partial y} (x * y) + z = x\).

Template Parameters

Tc	type of the summand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 57 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const fvar< T1 > &  x1, const fvar< T2 > &  x2, const fvar< T3 > &  x3  )

inline

The fused multiply-add operation (C99).

This double-based operation delegates to fma.

The function is defined by

fma(a, b, c) = (a * b) + c.

\[ \mbox{fma}(x, y, z) = \begin{cases} x\cdot y+z & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial z} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x1	First value.
x2	Second value.
x3	Third value.

Returns

Product of the first two values plus the third.

Definition at line 60 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const T1 &  x1, const fvar< T2 > &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 73 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const fvar< T1 > &  x1, const T2 &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 84 of file fma.hpp.

template<typename Ta , typename Tb , typename Tc , require_arithmetic_t< Tb > * = nullptr, require_all_var_t< Ta, Tc > * = nullptr>

var stan::math::fma ( Ta &&  x, Tb &&  y, Tc &&  z  )

inline

The fused multiply-add function for a variable, value, and variable (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + z = y\), and

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Template Parameters

Ta	type of the first multiplicand
Tb	type of the second multiplicand
Tc	type of the summand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 87 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const fvar< T1 > &  x1, const fvar< T2 > &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 95 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const T1 &  x1, const T2 &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 106 of file fma.hpp.

template<typename Tb , typename Tc , require_all_arithmetic_t< Tb, Tc > * = nullptr>

var stan::math::fma ( const var &  x, Tb &&  y, Tc &&  z  )

inline

The fused multiply-add function for a variable and two values (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The derivative is

\(\frac{d}{d x} (x * y) + z = y\).

Template Parameters

Tb	type of the second multiplicand
Tc	type of the summand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 116 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const fvar< T1 > &  x1, const T2 &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 116 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_all_stan_scalar_t< T1, T2, T3 > * = nullptr>

fvar<return_type_t<T1, T2, T3> > stan::math::fma ( const T1 &  x1, const fvar< T2 > &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 126 of file fma.hpp.

template<typename Ta , typename Tc , require_all_arithmetic_t< Ta, Tc > * = nullptr>

var stan::math::fma ( Ta &&  x, const var &  y, Tc &&  z  )

inline

The fused multiply-add function for a value, variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The derivative is

\(\frac{d}{d y} (x * y) + z = x\), and

Template Parameters

Ta	type of the first multiplicand
Tc	type of the summand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 139 of file fma.hpp.

template<typename Ta , typename Tb , require_all_arithmetic_t< Ta, Tb > * = nullptr>

var stan::math::fma ( Ta &&  x, Tb &&  y, const var &  z  )

inline

The fused multiply-add function for two values and a variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The derivative is

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Template Parameters

Ta	type of the first multiplicand
Tb	type of the second multiplicand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 162 of file fma.hpp.

template<typename Ta , require_arithmetic_t< Ta > * = nullptr>

var stan::math::fma ( Ta &&  x, const var &  y, const var &  z  )

inline

The fused multiply-add function for a value and two variables (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial y} (x * y) + z = x\), and

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Template Parameters

Ta	type of the first multiplicand

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 185 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 , require_any_matrix_t< T1, T2, T3 > * = nullptr, require_var_t< return_type_t< T1, T2, T3 >> * = nullptr>

auto stan::math::fma ( const T1 &  x, const T2 &  y, const T3 &  z  )

inline

The fused multiply-add function for three variables (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + z = y\), and

\(\frac{\partial}{\partial y} (x * y) + z = x\), and

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Parameters

x	First multiplicand.
y	Second multiplicand.
z	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 390 of file fma.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

double stan::math::fmax ( T1  x, T2  y  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x	First argument.
y	Second argument.

Returns

maximum of x or y and if one is NaN return the other

Definition at line 20 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 23 of file fmax.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::fmax ( T_a &&  a, T_b &&  b  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

a	First argument.
b	Second argument.

Returns

Maximum of a or b and if one is NaN return the other

Definition at line 31 of file fmax.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::fmax ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the fmax function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

fmax function applied to the two inputs.

Definition at line 36 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( double  x1, const fvar< T > &  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 46 of file fmax.hpp.

var stan::math::fmax ( const var &  a, const var &  b  )

inline

Returns the maximum of the two variable arguments (C99).

No new variable implementations are created, with this function defined as if by

fmax(a, b) = a if a's value is greater than b's, and .

fmax(a, b) = b if b's value is greater than or equal to a's.

\[ \mbox{fmax}(x, y) = \begin{cases} x & \mbox{if } x \geq y \\ y & \mbox{if } x < y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \geq y \\ 0 & \mbox{if } x < y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

If the first variable's value is larger than the second's, the first variable, otherwise the second variable.

Definition at line 61 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( const fvar< T > &  x1, double  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 69 of file fmax.hpp.

var stan::math::fmax ( const var &  a, double  b  )

inline

Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a's value is greater than b, then a is returned, otherwise a fresh variable implementation wrapping the value b is returned.

Parameters

a	First variable.
b	Second value

Returns

If the first variable's value is larger than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 91 of file fmax.hpp.

var stan::math::fmax ( double  a, const var &  b  )

inline

Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a is greater than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns

If the first value is larger than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 119 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmin ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file fmin.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

double stan::math::fmin ( T1  x, T2  y  )

inline

Return the lesser of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x	First argument.
y	Second argument.

Returns

Minimum of x or y and if one is NaN return the other

Definition at line 20 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmin ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 28 of file fmin.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::fmin ( T_a &&  a, T_b &&  b  )

inline

Return the lesser of the two specified arguments.

If one is not-a-number, return the other.

Parameters

a	First argument.
b	Second argument.

Returns

Minimum of a or b and if one is NaN return the other

Definition at line 31 of file fmin.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::fmin ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the fmin function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

fmin function applied to the two inputs.

Definition at line 36 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmin ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 42 of file fmin.hpp.

var stan::math::fmin ( const var &  a, const var &  b  )

inline

Returns the minimum of the two variable arguments (C99).

For fmin(a, b), if a's value is less than b's, then a is returned, otherwise b is returned.

\[ \mbox{fmin}(x, y) = \begin{cases} x & \mbox{if } x \leq y \\ y & \mbox{if } x > y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \leq y \\ 0 & \mbox{if } x > y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \leq y \\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

If the first variable's value is smaller than the second's, the first variable, otherwise the second variable.

Definition at line 57 of file fmin.hpp.

var stan::math::fmin ( const var &  a, double  b  )

inline

Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a's value is less than or equal to b, then a is returned, otherwise a fresh variable wrapping b is returned.

Parameters

a	First variable.
b	Second value

Returns

If the first variable's value is less than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 85 of file fmin.hpp.

var stan::math::fmin ( double  a, const var &  b  )

inline

Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a is less than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns

If the first value is smaller than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 113 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmod ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 15 of file fmod.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::fmod ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the fmod function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

fmod function applied to the two inputs.

Definition at line 22 of file fmod.hpp.

template<typename T >

fvar<T> stan::math::fmod ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 23 of file fmod.hpp.

template<typename T >

fvar<T> stan::math::fmod ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 33 of file fmod.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::fmod ( T_a &&  a, T_b &&  b  )

inline

Return the floating point remainder after dividing the first variable by the second (cmath).

The partial derivatives with respect to the variables are defined everywhere but where \(x = y\), but we set these to match other values, with

\(\frac{\partial}{\partial x} \mbox{fmod}(x, y) = 1\), and

\(\frac{\partial}{\partial y} \mbox{fmod}(x, y) = -\lfloor \frac{x}{y} \rfloor\).

\[ \mbox{fmod}(x, y) = \begin{cases} x - \lfloor \frac{x}{y}\rfloor y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial y} = \begin{cases} -\lfloor \frac{x}{y}\rfloor & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

Floating pointer remainder of dividing the first variable by the second.

Definition at line 66 of file fmod.hpp.

var stan::math::fmod ( const var &  a, const var &  b  )

inline

Return the floating point remainder after dividing the first variable by the second (cmath).

The partial derivatives with respect to the variables are defined everywhere but where \(x = y\), but we set these to match other values, with

\(\frac{\partial}{\partial x} \mbox{fmod}(x, y) = 1\), and

\(\frac{\partial}{\partial y} \mbox{fmod}(x, y) = -\lfloor \frac{x}{y} \rfloor\).

\[ \mbox{fmod}(x, y) = \begin{cases} x - \lfloor \frac{x}{y}\rfloor y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial y} = \begin{cases} -\lfloor \frac{x}{y}\rfloor & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

Floating pointer remainder of dividing the first variable by the second.

Definition at line 100 of file fmod.hpp.

var stan::math::fmod ( const var &  a, double  b  )

inline

Return the floating point remainder after dividing the the first variable by the second scalar (cmath).

The derivative with respect to the variable is

\(\frac{d}{d x} \mbox{fmod}(x, c) = \frac{1}{c}\).

Parameters

a	First variable.
b	Second scalar.

Returns

Floating pointer remainder of dividing the first variable by the second scalar.

Definition at line 117 of file fmod.hpp.

var stan::math::fmod ( double  a, const var &  b  )

inline

Return the floating point remainder after dividing the first scalar by the second variable (cmath).

The derivative with respect to the variable is

\(\frac{d}{d y} \mbox{fmod}(c, y) = -\lfloor \frac{c}{y} \rfloor\).

Parameters

a	First scalar.
b	Second variable.

Returns

Floating pointer remainder of dividing first scalar by the second variable.

Definition at line 134 of file fmod.hpp.

template<typename F , typename T >

constexpr auto stan::math::for_each ( F &&  f, T &&  t  )

inline

Apply a function to each element of a tuple.

Template Parameters

F	type with a valid operator()
T	Tuple

Parameters

f	A functor to apply over each element of the tuple.
t	A tuple

Definition at line 66 of file for_each.hpp.

template<typename F , typename T1 , typename T2 >

constexpr auto stan::math::for_each ( F &&  f, T1 &&  t1, T2 &&  t2  )

inline

Apply a function to each element of two tuples.

Template Parameters

F	type with a valid operator()
T1	Tuple
T2	Another tuple

Parameters

f	A functor to apply over each element of the tuple.
t1	A tuple
t2	Another tuple

Definition at line 82 of file for_each.hpp.

template<typename F , typename T1 , typename T2 , typename T3 >

constexpr auto stan::math::for_each ( F &&  f, T1 &&  t1, T2 &&  t2, T3 &&  t3  )

inline

Apply a function to each element of three tuples.

Template Parameters

F	type with a valid operator()
T1	Tuple
T2	Another tuple
T3	Another tuple

Parameters

f	A functor to apply over each element of the tuple.
t1	A tuple
t2	Another tuple
t3	Another tuple

Definition at line 104 of file for_each.hpp.

template<typename T_desired , typename T_actual , require_all_eigen_t< T_desired, T_actual > * = nullptr, std::enable_if_t< !std::is_same< value_type_t< T_actual >, value_type_t< T_desired >>::value||!internal::eigen_static_size_match(T_desired::RowsAtCompileTime, std::decay_t< T_actual >::RowsAtCompileTime)||!internal::eigen_static_size_match(T_desired::ColsAtCompileTime, std::decay_t< T_actual >::ColsAtCompileTime)> * = nullptr>

T_desired stan::math::forward_as ( const T_actual &  a )

inline

Assume which type we get.

If actual type is not convertible to assumed type or in case of eigen types compile time rows and columns are not the same and desired sizes are not dynamic this has return type of T_desired, but it only throws. This version should only be used where it is optimized away so the throw should never happen.

This is intended to be used in compile time branches that would otherwise trigger compile error even though they are never executed.

Template Parameters

T_desired	type of output we need to avoid compile time errors
T_actual	actual type of the argument

Parameters

a	input value

Returns

nothing, this always throws

Exceptions

always

throws std::runtime_error

Definition at line 122 of file forward_as.hpp.

template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_cdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 28 of file frechet_cdf.hpp.

template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 28 of file frechet_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 26 of file frechet_lcdf.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_shape, T_scale > * = nullptr>

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lpdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 31 of file frechet_lpdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  sigma  )

inline

Definition at line 101 of file frechet_lpdf.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

Eigen::Matrix<var, T::RowsAtCompileTime, T::ColsAtCompileTime> stan::math::from_var_value

(

const T &

a

)

Converts var_value into an Eigen Matrix.

Adjoint is propagated back to argument in the reverse pass.

Template Parameters

T	type of the input

Parameters

a	matrix to convert

Definition at line 20 of file from_var_value.hpp.

template<typename T , require_any_t< conjunction< is_eigen< T >, is_var< scalar_type_t< T >>>, std::is_same< std::decay_t< T >, var >, bool_constant<!std::is_same< scalar_type_t< T >, var >::value >> * = nullptr>

T stan::math::from_var_value

(

T &&

a

)

This is a no-op for Eigen containers of vars, scalars or prim types.

Template Parameters

T	type of the input

Parameters

a	matrix to convert

Definition at line 40 of file from_var_value.hpp.

template<typename T >

auto stan::math::from_var_value

(

const std::vector< T > &

a

)

Convert the elements of the std::vector input to var_value types if possible.

Template Parameters

T	type of elemnts of the input vector

Parameters

a	std::vector of elements to convert

Definition at line 52 of file from_var_value.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 25 of file gamma_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 25 of file gamma_lcdf.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lpdf ( const T_y &  y, const T_shape &  alpha, const T_inv_scale &  beta  )

inline

Definition at line 123 of file gamma_lpdf.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 16 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 35 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 50 of file gamma_p.hpp.

double stan::math::gamma_p ( double  z, double  a  )

inline

Return the value of the normalized, lower-incomplete gamma function applied to the specified argument.

This function is defined, including error conditions, as follows

\[ \mbox{gamma\_p}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ P(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ P(a, z)=\frac{1}{\Gamma(a)}\int_0^zt^{a-1}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_0^zt^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_0^z (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial z} = \frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Parameters

z	first argument
a	second argument

Returns

value of the normalized, lower-incomplete gamma function applied to z and a

Exceptions

std::domain_error

if either argument is not positive or if z is at a pole of the function

Definition at line 67 of file gamma_p.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::gamma_p ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the gamma_p function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

gamma_p function applied to the two inputs.

Definition at line 90 of file gamma_p.hpp.

var stan::math::gamma_p ( const var &  a, const var &  b  )

inline

Definition at line 102 of file gamma_p.hpp.

var stan::math::gamma_p ( const var &  a, double  b  )

inline

Definition at line 106 of file gamma_p.hpp.

var stan::math::gamma_p ( double  a, const var &  b  )

inline

Definition at line 110 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 46 of file gamma_q.hpp.

var stan::math::gamma_q ( const var &  a, const var &  b  )

inline

Definition at line 50 of file gamma_q.hpp.

var stan::math::gamma_q ( const var &  a, double  b  )

inline

Definition at line 54 of file gamma_q.hpp.

double stan::math::gamma_q ( double  x, double  a  )

inline

\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ Q(a, z)=\frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_z^\infty t^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_z^\infty (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial z} = -\frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Exceptions

domain_error

if x is at pole

Definition at line 55 of file gamma_q.hpp.

var stan::math::gamma_q ( double  a, const var &  b  )

inline

Definition at line 58 of file gamma_q.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::gamma_q ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the gamma_q function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

gamma_q function applied to the two inputs.

Definition at line 68 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 77 of file gamma_q.hpp.

template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >

return_type_t<T_y, T_F, T_G, T_V, T_W, T_m0, T_C0> stan::math::gaussian_dlm_obs_lpdf ( const T_y &  y, const T_F &  F, const T_G &  G, const T_V &  V, const T_W &  W, const T_m0 &  m0, const T_C0 &  C0  )

inline

Definition at line 288 of file gaussian_dlm_obs_lpdf.hpp.

template<class RNG >

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::gaussian_dlm_obs_rng ( const Eigen::MatrixXd &  F, const Eigen::MatrixXd &  G, const Eigen::VectorXd &  V, const Eigen::MatrixXd &  W, const Eigen::VectorXd &  m0, const Eigen::MatrixXd &  C0, const int  T, RNG &  rng  )

inline

Definition at line 142 of file gaussian_dlm_obs_rng.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::ColsAtCompileTime, EigMat::RowsAtCompileTime> stan::math::generalized_inverse ( const EigMat &  G )

inline

Returns the Moore-Penrose generalized inverse of the specified matrix.

The method is based on the Cholesky computation of the transform as specified in

• Courrieu, Pierre. 2008. Fast Computation of Moore-Penrose Inverse Matrices. arXiv 0804.4809

Template Parameters

EigMat

type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

G	specified matrix

Returns

Generalized inverse of the matrix (an empty matrix if the specified matrix has size zero).

Definition at line 33 of file generalized_inverse.hpp.

template<typename VarMat , require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::generalized_inverse ( const VarMat &  G )

inline

Definition at line 64 of file generalized_inverse.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< T > &  x, size_t  i, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i	Index into vector plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at i - 1

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 28 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< T > > &  x, size_t  i1, size_t  i2, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 51 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< T > > > &  x, size_t  i1, size_t  i2, size_t  i3, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 75 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< T > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 101 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 129 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 159 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 190 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, size_t  i8, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
i8	Eigth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 222 of file get_base1.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vector_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, 1, Eigen::Dynamic> stan::math::get_base1 ( const EigMat &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a copy of the row of the specified matrix at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Template Parameters

EigMat

type of the matrix

Parameters

x	Matrix from which to get a row
m	Index into matrix plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Row of matrix at i - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 253 of file get_base1.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

const value_type_t<EigMat>& stan::math::get_base1 ( const EigMat &  x, size_t  m, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

EigMat

type of the matrix

Parameters

x	Matrix from which to get a row
m	Row index plus 1.
n	Column index plus 1.
error_msg	Error message if either index is out of range.
idx	Nested index level to report in error message if either index is out of range.

Returns

Value of matrix at row m - 1 and column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 277 of file get_base1.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

const value_type_t<EigVec>& stan::math::get_base1 ( const EigVec &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified Eigen vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

EigVec

type of the vector

Parameters

x	Eigen vector from which to get a value.
m	Row index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of Eigen vector at row m - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 301 of file get_base1.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< T > &  x, size_t  i, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i	Index into vector plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at i - 1

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 27 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< T > > &  x, size_t  i1, size_t  i2, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 50 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< T > > > &  x, size_t  i1, size_t  i2, size_t  i3, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 74 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< T > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 100 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 127 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 156 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 187 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, size_t  i8, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
i8	Eigth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 219 of file get_base1_lhs.hpp.

template<typename T >

Eigen::Block<Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> > stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a copy of the row of the specified vector at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Index into matrix plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Row of matrix at i - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 251 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Row index plus 1.
n	Column index plus 1.
error_msg	Error message if either index is out of range.
idx	Nested index level to report in error message if either index is out of range.

Returns

Value of matrix at row m - 1 and column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 275 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified column vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Column vector from which to get a value.
m	Row index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of column vector at row m - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 298 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, 1, Eigen::Dynamic > &  x, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified row vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Row vector from which to get a value.
n	Column index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of row vector at column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 320 of file get_base1_lhs.hpp.

template<typename T >

T stan::math::get_imag

(

const std::complex< T > &

z

)

Return the imaginary component of the complex argument.

Template Parameters

T	value type of complex argument

Parameters

[in]

z

complex value whose imaginary component is extracted

Returns

imaginary component of argument

Definition at line 18 of file get_imag.hpp.

template<typename T_lp , typename T_lp_accum >

return_type_t<T_lp, T_lp_accum> stan::math::get_lp ( const T_lp &  lp, const accumulator< T_lp_accum > &  lp_accum  )

inline

Definition at line 10 of file get_lp.hpp.

template<typename T >

T stan::math::get_real

(

const std::complex< T > &

z

)

Return the real component of the complex argument.

Template Parameters

T	value type of complex argument

Parameters

[in]

z

complex value whose real component is extracted

Returns

real component of argument

Definition at line 18 of file get_real.hpp.

template<typename T_x , typename T_sigma >

Eigen::Matrix<return_type_t<T_x, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &	x,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x	type of std::vector of elements
T_sigma	type of sigma

Parameters

x	std::vector of elements that can be used in dot product. This function assumes each element of x is the same size.
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is positive semi-definite

Exceptions

std::domain_error

if sigma < 0, nan, inf or x is nan or infinite

Definition at line 38 of file gp_dot_prod_cov.hpp.

template<typename T_x , typename T_sigma >

Eigen::Matrix<return_type_t<T_x, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< T_x > &	x,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x	type of std::vector of double
T_sigma	type of sigma

Parameters

x	std::vector of elements that can be used in transpose and multiply This function assumes each element of x is the same size.
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is positive semi-definite

Exceptions

std::domain_error

if sigma < 0, nan, inf or x is nan or infinite

Definition at line 100 of file gp_dot_prod_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &	x1,
		const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &	x2,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix of differing x's.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x1	type of first std::vector of elements
T_x2	type of second std::vector of elements
T_sigma	type of sigma

Parameters

x1	std::vector of elements that can be used in dot_product
x2	std::vector of elements that can be used in dot_product
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is not always symmetric

Exceptions

std::domain_error

if sigma < 0, nan or inf or if x1 or x2 are nan or inf

Definition at line 157 of file gp_dot_prod_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< T_x1 > &	x1,
		const std::vector< T_x2 > &	x2,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix of differing x's.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x1	type of first std::vector of double
T_x2	type of second std::vector of double
T_sigma	type of sigma

Parameters

x1	std::vector of elements that can be used in dot_product
x2	std::vector of elements that can be used in dot_product
sigma	is the constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is not always symmetric

Exceptions

std::domain_error

if sigma < 0, nan or inf or if x1 or x2 are nan or inf

Definition at line 221 of file gp_dot_prod_cov.hpp.

template<typename T_x , typename T_sigma , require_st_arithmetic< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>

Eigen::Matrix<var, -1, -1> stan::math::gp_exp_quad_cov ( const std::vector< T_x > &  x, const T_sigma  sigma, const var  length_scale  )

inline

Returns a squared exponential kernel.

Template Parameters

T_x	type of elements in the vector

Parameters

x	std::vector input that can be used in square distance Assumes each element of x is the same size
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 33 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Returns a squared exponential kernel.

Template Parameters

T_x	type for each scalar
T_sigma	type of parameter sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in square distance. This function assumes each element of x is the same size.
sigma	marginal standard deviation or magnitude
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 126 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a squared exponential kernel.

Template Parameters

T_x	type for each scalar
T_sigma	type of parameter sigma
T_l	type of each length scale parameter

Parameters

x	std::vector of Eigen vectors of scalars.
sigma	marginal standard deviation or magnitude
length_scale	std::vector length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 166 of file gp_exp_quad_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute posterior predictive density.

Template Parameters

T_x1	type of first std::vector of scalars
T_x2	type of second std::vector of scalars This function assumes each element of x1 and x2 are the same size.
T_sigma	type of sigma
T_l	type of of length scale

Parameters

x1	std::vector of elements that can be used in square distance
x2	std::vector of elements that can be used in square distance
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 209 of file gp_exp_quad_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Template Parameters

T_x1	type of first std::vector of elements
T_x2	type of second std::vector of elements
T_s	type of sigma
T_l	type of length scale

Parameters

x1	std::vector of Eigen vectors of scalars.
x2	std::vector of Eigen vectors of scalars.
sigma	standard deviation
length_scale	std::vector of length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 258 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern exponential covariance Matrix.

\[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')^2}{2l^2}) \]

where d(x, x') is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of parameter sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in stan::math::distance
sigma	standard deviation or magnitude
length_scale	length scale

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 38 of file gp_exponential_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern exponential covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \]

where d(x, x') is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type for each parameter sigma
T_l	type for each length scale parameter

Parameters

x	std::vector of Eigen::vectors of scalars
sigma	standard deviation that can be used in stan::math::square
length_scale	std::vector of length scales

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 103 of file gp_exponential_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern exponential cross covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')}{l}) \]

where d(x, x') is the Euclidean distance This function is for the cross covariance matrix needed to compute the posterior predictive distribution

Template Parameters

T_x1	first type of scalars contained in vector x1
T_x2	second type of scalars contained in vector x2
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of scalars that can be used in squared_distance
x2	std::vector of scalars that can be used in squared_distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 171 of file gp_exponential_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern exponential cross covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \]

where \(d(x, x')\) is the Euclidean distance

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Template Parameters

T_x1	first type of std::vector of scalars
T_x2	second type of std::vector of scalars
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of Eigen vectors of scalars
x2	std::vector of Eigen vectors of scalars
length_scale	parameter length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 248 of file gp_exponential_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 3/2 covariance matrix.

\[ k(x, x') = \sigma^2(1 + \frac{\sqrt{3}d(x, x')}{l})exp(-\frac{\sqrt{3}d(x, x')}{l}) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of parameter of sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in squared distance
length_scale	length scale
sigma	marginal standard deviation or magnitude

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 39 of file gp_matern32_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 3/2 covariance matrix.

\[ k(x, x') = \sigma^2(1 + \sqrt{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) exp(-\sqrt{3}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) \]

where \(d(x, x')\) is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of element of parameter sigma
T_l	type of each length scale parameter

Parameters

x	std::vector of Eigen vectors of scalars.
length_scale	std::vector of length scales
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 109 of file gp_matern32_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 3/2 cross covariance matrix.

\[ k(x, x') = \sigma^2(1 + \frac{\sqrt{3}d(x, x')}{l})exp(-\sqrt{3}\frac{d(x, x')}{l}) \]

where \(d(x, x')\) is the Euclidean distance.

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Template Parameters

T_x1	first type of scalars contained in vector x1
T_x2	second type of scalars contained in vector x2
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of scalars that can be used in squared_distance
x2	std::vector of scalars that can be used in squared_distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 186 of file gp_matern32_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 3/2 cross covariance matrix.

\[ k(x, x') = \sigma^2(1 + \sqrt{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) exp(-\sqrt{3}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) \]

where \(d(x, x')\) is the Euclidean distance

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Template Parameters

T_x1	first type of std::vector of scalars
T_x2	second type of std::vector of scalars
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of Eigen vectors of scalars
x2	std::vector of Eigen vectors of scalars
length_scale	parameter length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 268 of file gp_matern32_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with one input vector.

\[ k(x, x') = \sigma^2\bigg(1 + \frac{\sqrt{5}d(x, x')}{l} + \frac{5d(x, x')^2}{3l^2}\bigg) exp\bigg(-\frac{5 d(x, x')}{l}\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type of elements contained in vector x
T_s	type of element of sigma, the magnitude
T_l	type of elements of length scale

Parameters

x	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 41 of file gp_matern52_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with one input vector with automatic relevance determination (ARD).

\[ k(x, x') = \sigma^2\bigg(1 + \sqrt{5}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}} + \frac{5}{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}\bigg) exp\bigg(-\frac{5}{3}\bigg(\sqrt{\sum_{k=1}^K{\frac{d(x, x')^2}{l_k^2}} }\bigg)\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type of elements contained in vector x
T_s	type of element of sigma, the magnitude
T_l	type of elements in vector of length scale

Parameters

x	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain	error if sigma <= 0, l <= 0, or x is nan or inf
std::invalid_argument	if length scale size != dimension of x

Definition at line 112 of file gp_matern52_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with two different input vectors.

\[ k(x, x') = \sigma^2\bigg(1 + \frac{\sqrt{5}d(x, x')}{l} + \frac{5d(x, x')^2}{3l^2}\bigg) exp\bigg(-\frac{5 d(x, x')}{l}\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x1	type of elements contained in vector x1
T_x2	type of elements contained in vector x2
T_s	type of element of sigma, the magnitude
T_l	type of elements of length scale

Parameters

x1	std::vector of elements that can be used in stan::math::distance
x2	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 188 of file gp_matern52_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with two input vectors with automatic relevance determination (ARD).

\[ k(x, x') = \sigma^2\bigg(1 + \sqrt{5}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}} + \frac{5}{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}\bigg) exp\bigg(-\frac{5}{3}\bigg(\sqrt{\sum_{k=1}^K{\frac{d(x, x')^2}{l_k^2}} }\bigg)\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x1	type of elements contained in vector x1
T_x2	type of elements contained in vector x2
T_s	type of element of sigma, the magnitude
T_l	type of elements in vector of length scale

Parameters

x1	std::vector of elements that can be used in stan::math::distance
x2	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain	error if sigma <= 0, l <= 0, or x is nan or inf
std::invalid_argument	if length scale size != dimension of x1 or x2

Definition at line 270 of file gp_matern52_cov.hpp.

template<typename T_x , typename T_sigma , require_st_arithmetic< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>

Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_periodic_cov ( const std::vector< T_x > &  x, const T_sigma  sigma, const var  l, const var  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_i,\mathbf{X}_j), \) where \( \mathbf{X}_i \) is the \(i\)-th row of \( \mathbf{X} \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Template Parameters

T_x	type of elements in the std::vector

Parameters

x	std::vector of input elements. Assumes that all elements of x have the same size.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0, or x is nan or infinite

Definition at line 42 of file gp_periodic_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l , typename T_p >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l, T_p>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_periodic_cov ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  l, const T_p &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_i,\mathbf{X}_j), \) where \( \mathbf{X}_i \) is the \(i\)-th row of \( \mathbf{X} \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Template Parameters

T_x	type of std::vector elements of x. T_x can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_sigma	type of sigma
T_l	type of length-scale
T_p	type of period

Parameters

x	std::vector of input elements. This function assumes that all elements of x have the same size.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0 or x is nan or infinite

Definition at line 47 of file gp_periodic_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l , typename T_p >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l, T_p>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_periodic_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  l, const T_p &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using inputs \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_{1_i},\mathbf{X}_{2_j}), \) where \( \mathbf{X}_{1_i} \) and \( \mathbf{X}_{2_j} \) are the \(i\)-th and \(j\)-th rows of \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Template Parameters

T_x1	type of std::vector elements of x1 T_x1 can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_x2	type of std::vector elements of x2 T_x2 can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_sigma	type of sigma
T_l	type of length-scale
T_p	type of period

Parameters

x1	std::vector of first input elements
x2	std::vector of second input elements. This function assumes that all the elements of x1 and x2 have the same sizes.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0 , x1 or x2 is nan or infinite

Definition at line 127 of file gp_periodic_cov.hpp.

template<typename Vari >

static void stan::math::grad ( Vari *  vi )

static

void stan::math::grad ( var &  v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &  x, Eigen::VectorXd &  g  )

inline

Propagate chain rule to calculate gradients starting from the specified variable.

Resizes the input vector to be the correct size.

The grad() function does not itself recover any memory. use recover_memory() or recover_memory_nested() to recover memory.

Parameters

[in]	v	Value of function being differentiated
[in]	x	Variables being differentiated with respect to
[out]	g	Gradient, d/dx v, evaluated at x.

Definition at line 24 of file grad.hpp.

static void stan::math::grad ( )

static

Compute the gradient for all variables starting from the end of the AD tape.

This function does not recover memory. The chain rule is applied working down the stack from the last vari created on the AD tape and then calling each vari's chain() method in turn.

This function computes a nested gradient only going back as far as the last nesting.

This function does not recover any memory from the computation.

Definition at line 26 of file grad.hpp.

template<typename Vari >

static void stan::math::grad ( Vari *  vi )

static

Compute the gradient for all variables starting from the specified root variable implementation.

Does not recover memory. This chainable variable's adjoint is initialized using the method init_dependent() and then the chain rule is applied working down the stack from this vari and calling each vari's chain() method in turn.

This function computes a nested gradient only going back as far as the last nesting.

This function does not recover any memory from the computation.

Parameters

vi	Variable implementation for root of partial derivative propagation.

Definition at line 51 of file grad.hpp.

template<typename T >

void stan::math::grad_2F1	(	T &	g_a1,
		T &	g_b1,
		const T &	a1,
		const T &	a2,
		const T &	b1,
		const T &	z,
		double	precision = 1e-14,
		int	max_steps = 1e6
	)

Gradients of the hypergeometric function, 2F1.

Calculate the gradients of the hypergeometric function (2F1) as the power series stopping when the series converges to within precision or throwing when the function takes max_steps steps.

This power-series representation converges for all gradients under the same conditions as the 2F1 function itself.

Template Parameters

T	type of arguments and result

Parameters

[out]	g_a1	g_a1 reference to gradient of 2F1 w.r.t. a1, result.
[out]	g_b1	g_b1 reference to gradient of 2F1 w.r.t. b1, result.
[in]	a1	a1 see generalized hypergeometric function definition.
[in]	a2	a2 see generalized hypergeometric function definition.
[in]	b1	b1 see generalized hypergeometric function definition.
[in]	z	z see generalized hypergeometric function definition.
[in]	precision	magnitude of the increment of the infinite sum to truncate the sum at.
[in]	max_steps	number of steps to take.

Definition at line 39 of file grad_2F1.hpp.

template<typename T >

void stan::math::grad_F32	(	T *	g,
		const T &	a1,
		const T &	a2,
		const T &	a3,
		const T &	b1,
		const T &	b2,
		const T &	z,
		const T &	precision = 1e-6,
		int	max_steps = 1e5
	)

Gradients of the hypergeometric function, 3F2.

Calculate the gradients of the hypergeometric function (3F2) as the power series stopping when the series converges to within precision or throwing when the function takes max_steps steps.

This power-series representation converges for all gradients under the same conditions as the 3F2 function itself.

Template Parameters

T	type of arguments and result

Parameters

[out]	g	g pointer to array of six values of type T, result.
[in]	a1	a1 see generalized hypergeometric function definition.
[in]	a2	a2 see generalized hypergeometric function definition.
[in]	a3	a3 see generalized hypergeometric function definition.
[in]	b1	b1 see generalized hypergeometric function definition.
[in]	b2	b2 see generalized hypergeometric function definition.
[in]	z	z see generalized hypergeometric function definition.
[in]	precision	precision of the infinite sum
[in]	max_steps	number of steps to take

Definition at line 39 of file grad_F32.hpp.

template<typename F >

void stan::math::grad_hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	H,
		std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &	grad_H
	)

Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument.

The functor must implement

fvar<fvar<var> > operator()(const Eigen::Matrix<fvar<fvar<var> >, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar and var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	H	Hessian of function at argument
[out]	grad_H	Gradient of the Hessian of function at argument

Definition at line 42 of file grad_hessian.hpp.

void stan::math::grad_inc_beta ( double &  g1, double &  g2, double  a, double  b, double  z  )

inline

Definition at line 22 of file grad_inc_beta.hpp.

template<typename T >

void stan::math::grad_inc_beta	(	fvar< T > &	g1,
		fvar< T > &	g2,
		fvar< T >	a,
		fvar< T >	b,
		fvar< T >	z
	)

Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments.

Uses the equivalence to a hypergeometric function. See http://dlmf.nist.gov/8.17#ii

Template Parameters

T	inner type of the fvar

Parameters

[out]	g1	d/da
[out]	g2	d/db
[in]	a	a
[in]	b	b
[in]	z	z

Definition at line 34 of file grad_inc_beta.hpp.

void stan::math::grad_inc_beta ( var &  g1, var &  g2, const var &  a, const var &  b, const var &  z  )

inline

Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments.

Uses the equivalence to a hypergeometric function. See http://dlmf.nist.gov/8.17#ii

Parameters

[out]	g1	d/da
[out]	g2	d/db
[in]	a	a
[in]	b	b
[in]	z	z

Definition at line 36 of file grad_inc_beta.hpp.

template<typename T >

void stan::math::grad_reg_inc_beta	(	T &	g1,
		T &	g2,
		const T &	a,
		const T &	b,
		const T &	z,
		const T &	digammaA,
		const T &	digammaB,
		const T &	digammaSum,
		const T &	betaAB
	)

Computes the gradients of the regularized incomplete beta function.

Specifically, this function computes gradients of ibeta(a, b, z), with respect to the arguments a and b.

Template Parameters

T	type of arguments

Parameters

[out]	g1	partial derivative of ibeta(a, b, z) with respect to a
[out]	g2	partial derivative of ibeta(a, b, z) with respect to b
[in]	a	a
[in]	b	b
[in]	z	z
[in]	digammaA	the value of digamma(a)
[in]	digammaB	the value of digamma(b)
[in]	digammaSum	the value of digamma(a + b)
[in]	betaAB	the value of beta(a, b)

Definition at line 33 of file grad_reg_inc_beta.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::grad_reg_inc_gamma	(	T1	a,
		T2	z,
		T1	g,
		T1	dig,
		double	precision = 1e-6,
		int	max_steps = 1e5
	)

Gradient of the regularized incomplete gamma functions igamma(a, z)

For small z, the gradient is computed via the series expansion; for large z, the series is numerically inaccurate due to cancellation and the asymptotic expansion is used.

Template Parameters

T1	type of the shape parameter
T2	type of the location parameter

Parameters

a	shape parameter, a > 0
z	location z >= 0
g	stan::math::tgamma(a) (precomputed value)
dig	boost::math::digamma(a) (precomputed value)
precision	required precision; applies to series expansion only
max_steps	number of steps to take.

Exceptions

throws

std::domain_error if not converged after max_steps or increment overflows to inf.

For the asymptotic expansion, the gradient is given by:

\[ \begin{array}{rcl} \Gamma(a, z) & = & z^{a-1}e^{-z} \sum_{k=0}^N \frac{(a-1)_k}{z^k} \qquad , z \gg a\\ Q(a, z) & = & \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \frac{(a-1)_k}{z^k}\\ (a)_k & = & (a)_{k-1}(a-k)\\ \frac{d}{da} (a)_k & = & (a)_{k-1} + (a-k)\frac{d}{da} (a)_{k-1}\\ \frac{d}{da}Q(a, z) & = & (log(z) - \psi(a)) Q(a, z)\\ && + \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \left(\frac{d}{da} (a-1)_k\right) \frac{1}{z^k} \end{array} \]

Definition at line 52 of file grad_reg_inc_gamma.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::grad_reg_lower_inc_gamma	(	const T1 &	a,
		const T2 &	z,
		double	precision = 1e-10,
		int	max_steps = 1e5
	)

Computes the gradient of the lower regularized incomplete gamma function.

The lower incomplete gamma function derivative w.r.t its first parameter (a) seems to have no standard source. It also appears to have no widely known approximate implementation. Gautschi (1979) has a thorough discussion of the calculation of the lower regularized incomplete gamma function itself and some stability issues.

Reference: Gautschi, Walter (1979) ACM Transactions on mathematical software. 5(4):466-481

We implemented calculations for d(gamma_p)/da by taking derivatives of formulas suggested by Gauschi and others and testing them against an outside source (Mathematica). We took three implementations which can cover the range {a:[0,20], z:[0,30]} with absolute error < 1e-10 with the exception of values near (0,0) where the error is near 1e-5. Relative error is also <<1e-6 except for regions where the gradient approaches zero.

Gautschi suggests calculating the lower incomplete gamma function for small to moderate values of $z$ using the approximation:

\[ \frac{\gamma(a,z)}{\Gamma(a)}=z^a e^-z \sum_n=0^\infty \frac{z^n}{\Gamma(a+n+1)} \]

We write the derivative in the form:

\[ \frac{d\gamma(a,z)\Gamma(a)}{da} = \frac{\log z}{e^z} \sum_n=0^\infty \frac{z^{a+n}}{\Gamma(a+n+1)} - \frac{1}{e^z} \sum_n=0^\infty \frac{z^{a+n}}{\Gamma(a+n+1)}\psi^0(a+n+1) \]

This calculation is sufficiently accurate for small $a$ and small $z$. For larger values and $a$ and $z$ we use it in its log form:

\[ \frac{d \gamma(a,z)\Gamma(a)}{da} = \frac{\log z}{e^z} \sum_n=0^\infty \exp[(a+n)\log z - \log\Gamma(a+n+1)] - \sum_n=0^\infty \exp[(a+n)\log z - \log\Gamma(a+n+1) + \log\psi^0(a+n+1)] \]

For large $z$, Gauschi recommends using the upper incomplete Gamma instead and the negative of its derivative turns out to be more stable and accurate for larger $z$ and for some combinations of $a$ and $z$. This is a log-scale implementation of the derivative of the formulation suggested by Gauschi (1979). For some values it defers to the negative of the gradient for the gamma_q function. This is based on the suggestion by Gauschi (1979) that for large values of $z$ it is better to carry out calculations using the upper incomplete Gamma function.

Branching for choice of implementation for the lower incomplete regularized gamma function gradient. The derivative based on Gautschi's formulation appears to be sufficiently accurate everywhere except for large z and small to moderate a. The intersection between the two regions is a radius 12 quarter circle centered at a=0, z=30 although both implementations are satisfactory near the intersection.

Some limits that could be treated, e.g., infinite z should return tgamma(a) * digamma(a), throw instead to match the behavior of, e.g., boost::math::gamma_p

Template Parameters

T1	type of a
T2	type of z

Parameters

[in]	a	shared with complete Gamma
[in]	z	value to integrate up to
[in]	precision	series terminates when increment falls below this value.
[in]	max_steps	number of terms to sum before throwing

Exceptions

std::domain_error

if the series does not converge to requested precision before max_steps.

Definition at line 110 of file grad_reg_lower_inc_gamma.hpp.

template<typename F >

void stan::math::grad_tr_mat_times_hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	M,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_tr_MH
	)

Definition at line 15 of file grad_tr_mat_times_hessian.hpp.

template<typename T , typename F >

void stan::math::gradient	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

fvar<T> operator()(const Eigen::Matrix<T, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters

T	type of the elements in the vector
F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument

Definition at line 40 of file gradient.hpp.

template<typename F >

void stan::math::gradient	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

var operator()(const Eigen::Matrix<var, Eigen::Dynamic, 1>&)

using only operations that are defined for var. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument

Definition at line 42 of file gradient.hpp.

template<typename T1 , typename T2 , typename F >

void stan::math::gradient_dot_vector	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	v,
		T1 &	fx,
		T1 &	grad_fx_dot_v
	)

Definition at line 13 of file gradient_dot_vector.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::gumbel_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  beta  )

inline

Definition at line 105 of file gumbel_lpdf.hpp.

template<typename T , require_vector_t< T > * = nullptr>

auto stan::math::head ( const T &  v, size_t  n  )

inline

Return the specified number of elements as a vector or row vector (same as input) from the front of the specified vector or row vector.

Template Parameters

T	type of the vector

Parameters

v	Vector input.
n	Size of return.

Returns

The first n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 22 of file head.hpp.

template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>

auto stan::math::head ( T_x &&  x, size_t  n  )

inline

Return the specified number of elements as a vector or row vector (same as input) from the front of the specified vector or row vector.

Template Parameters

T_x	type of input kernel generator expression.

Parameters

x	input kernel generator expression.
n	Size of return.

Returns

The first n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 24 of file head.hpp.

template<typename T >

std::vector<T> stan::math::head	(	const std::vector< T > &	sv,
		size_t	n
	)

Return the specified number of elements as a standard vector from the front of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

sv	Standard vector.
n	Size of return.

Returns

The first n elements of sv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 40 of file head.hpp.

template<typename T , typename F >

void stan::math::hessian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in time O(N^3) time and O(N^2) space.

The advantage over the mixed definition, which is faster for Hessians, is that this version is itself differentiable.

The functor must implement

fvar<fvar<T> > operator()(const Eigen::Matrix<fvar<fvar<T> >, Eigen::Dynamic, 1>&)

using only operations that are defined for the argument type.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

T	type of elements in the vector and matrix
F	type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad	gradient of function at argument
[out]	H	Hessian of function at argument

Definition at line 41 of file hessian.hpp.

template<typename F >

void stan::math::hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space.

The functor must implement

fvar<var> operator()(const Eigen::Matrix<fvar<var>, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar and var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad	gradient of function at argument
[out]	H	Hessian of function at argument

Definition at line 42 of file hessian.hpp.

template<typename F >

void stan::math::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	v,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 14 of file hessian_times_vector.hpp.

template<typename T , typename F >

void stan::math::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	v,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 39 of file hessian_times_vector.hpp.

template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>

void stan::math::hmm_check ( const T_omega &  log_omegas, const T_Gamma &  Gamma, const T_rho &  rho, const char *  function  )

inline

Check arguments for hidden Markov model functions with a discrete latent state (lpdf, rng for latent states, and marginal probabilities for latent sates).

Template Parameters

T_omega	type of the log likelihood matrix
T_Gamma	type of the transition matrix
T_rho	type of the initial guess vector

Parameters

[in]	log_omegas	log matrix of observational densities.
[in]	Gamma	transition density between hidden states.
[in]	rho	initial state
[in]	function	the name of the function using the arguments.

Exceptions

<tt>std::invalid_argument</tt>	if Gamma is not square or if the size of rho is not the number of rows of log_omegas.
<tt>std::domain_error</tt>	if rho is not a simplex or the rows of Gamma are not a simplex.

Definition at line 30 of file hmm_check.hpp.

template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>

Eigen::MatrixXd stan::math::hmm_hidden_state_prob ( const T_omega &  log_omegas, const T_Gamma &  Gamma, const T_rho &  rho  )

inline

For a hidden Markov model with observation y, hidden state x, and parameters theta, compute the marginal probability vector for each x, given y and theta, p(x_i | y, theta).

In this setting, the hidden states are discrete and take values over the finite space {1, ..., K}. Hence for each hidden variable x, we compute a simplex with K elements. The final result is stored in a K by N matrix, where N is the length of x. log_omegas is a matrix of observational densities, where the (i, j)th entry corresponds to the density of the ith observation, y_i, given x_i = j. The transition matrix Gamma is such that the (i, j)th entry is the probability that x_n = j given x_{n - 1} = i. The rows of Gamma are simplexes. This function cannot be used to reconstruct the marginal distributon of a state sequence given parameters and an observation sequence, p(x | y, theta), because it only computes marginals on a state-by-state basis.

Template Parameters

T_omega	type of the log likelihood matrix
T_Gamma	type of the transition matrix
T_rho	type of the initial guess vector

Parameters

[in]	log_omegas	log matrix of observational densities
[in]	Gamma	transition density between hidden states
[in]	rho	initial state

Returns

the posterior probability for each latent state

Exceptions

<tt>std::invalid_argument</tt>	if Gamma is not square or if the size of rho is not the number of rows of log_omegas
<tt>std::domain_error</tt>	if rho is not a simplex and of the rows of Gamma are not a simplex

Definition at line 48 of file hmm_hidden_state_prob.hpp.

template<typename T_omega , typename T_Gamma , typename T_rho , class RNG , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>

std::vector<int> stan::math::hmm_latent_rng ( const T_omega &  log_omegas, const T_Gamma &  Gamma, const T_rho &  rho, RNG &  rng  )

inline

For a hidden Markov model with observation y, hidden state x, and parameters theta, generate samples from the posterior distribution of the hidden states, x.

In this setting, the hidden states are discrete and takes values over the finite space {1, ..., K}. log_omegas is a matrix of observational densities, where the (i, j)th entry corresponds to the density of the ith observation, y_i, given x_i = j. The transition matrix Gamma is such that the (i, j)th entry is the probability that x_n = j given x_{n - 1} = i. The rows of Gamma are simplexes.

Template Parameters

T_omega	type of the log likelihood matrix
T_Gamma	type of the transition matrix
T_rho	type of the initial guess vector

Parameters

[in]	log_omegas	log matrix of observational densities.
[in]	Gamma	transition density between hidden states.
[in]	rho	initial state
[in]	rng	random number generator

Returns

sample from the posterior distribution of the hidden states.

Exceptions

<tt>std::invalid_argument</tt>	if Gamma is not square, when we have at least one transition, or if the size of rho is not the number of rows of log_omegas.
<tt>std::domain_error</tt>	if rho is not a simplex and of the rows of Gamma are not a simplex (when there is at least one transition).

Definition at line 44 of file hmm_latent_rng.hpp.

template<typename T_omega , typename T_Gamma , typename T_rho , require_all_eigen_t< T_omega, T_Gamma > * = nullptr, require_eigen_col_vector_t< T_rho > * = nullptr>

auto stan::math::hmm_marginal ( const T_omega &  log_omegas, const T_Gamma &  Gamma, const T_rho &  rho  )

inline

For a Hidden Markov Model with observation y, hidden state x, and parameters theta, return the log marginal density, log p(y | theta).

In this setting, the hidden states are discrete and take values over the finite space {1, ..., K}. The marginal lpdf is obtained via a forward pass, and the derivative is calculated with an adjoint method, e.g (Betancourt, Margossian, & Leos-Barajas, 2020). log_omegas is a matrix of observational densities, where the (i, j)th entry corresponds to the density of the jth observation, y_j, given x_j = i. The transition matrix Gamma is such that the (i, j)th entry is the probability that x_n = j given x_{n - 1} = i. The rows of Gamma are simplexes.

Template Parameters

T_omega	type of the log likelihood matrix
T_Gamma	type of the transition matrix
T_rho	type of the initial guess vector

Parameters

[in]	log_omegas	log matrix of observational densities.
[in]	Gamma	transition density between hidden states.
[in]	rho	initial state

Returns

log marginal density.

Exceptions

<tt>std::invalid_argument</tt>	if Gamma is not square, when we have at least one transition, or if the size of rho is not the number of rows of log_omegas.
<tt>std::domain_error</tt>	if rho is not a simplex and of the rows of Gamma are not a simplex (when there is at least one transition).

Definition at line 77 of file hmm_marginal.hpp.

template<typename T_omega , typename T_Gamma , typename T_rho , typename T_alpha >

auto stan::math::hmm_marginal_val ( const Eigen::Matrix< T_omega, Eigen::Dynamic, Eigen::Dynamic > &  omegas, const T_Gamma &  Gamma_val, const T_rho &  rho_val, Eigen::Matrix< T_alpha, Eigen::Dynamic, Eigen::Dynamic > &  alphas, Eigen::Matrix< T_alpha, Eigen::Dynamic, 1 > &  alpha_log_norms, T_alpha &  norm_norm  )

inline

Definition at line 20 of file hmm_marginal.hpp.

template<typename T , typename... Ptrs, std::enable_if_t< sizeof...(Ptrs) > = 1>

Ptrs stan::math::holder	(	T &&	arg,
		Ptrs *...	pointers
	)

Definition at line 240 of file holder.hpp.

template<typename T >

T stan::math::holder

(

T &&

arg

)

Definition at line 245 of file holder.hpp.

template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >

double stan::math::hypergeometric_lpmf	(	const T_n &	n,
		const T_N &	N,
		const T_a &	a,
		const T_b &	b
	)

Definition at line 18 of file hypergeometric_lpmf.hpp.

template<typename T_n , typename T_N , typename T_a , typename T_b >

double stan::math::hypergeometric_lpmf ( const T_n &  n, const T_N &  N, const T_a &  a, const T_b &  b  )

inline

Definition at line 59 of file hypergeometric_lpmf.hpp.

template<class RNG >

int stan::math::hypergeometric_rng ( int  N, int  a, int  b, RNG &  rng  )

inline

Definition at line 14 of file hypergeometric_rng.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

double stan::math::hypot ( T1  x, T2  y  )

inline

Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are x and y, the result is sqrt(x * x + y * y).

Parameters

x	First argument.
y	Second argument.

Returns

Length of hypotenuse of right triangle with opposite and adjacent side lengths x and y.

Definition at line 24 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypotenuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 26 of file hypot.hpp.

var stan::math::hypot ( const var &  a, const var &  b  )

inline

Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99).

The partial derivatives are given by

\(\frac{\partial}{\partial x} \sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}}\), and

\(\frac{\partial}{\partial y} \sqrt{x^2 + y^2} = \frac{y}{\sqrt{x^2 + y^2}}\).

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypotenuse.

Definition at line 27 of file hypot.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::hypot ( T_a &&  a, T_b &&  b  )

inline

Returns the elementwise hypot() of the input.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Elementwise hypot() of the input.

Definition at line 28 of file hypot.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

auto stan::math::hypot ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the hypot function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

hypot function applied to the two inputs.

Definition at line 42 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( const fvar< T > &  x1, double  x2  )

inline

Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypotenuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 46 of file hypot.hpp.

var stan::math::hypot ( const var &  a, double  b  )

inline

Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99).

The derivative is

\(\frac{d}{d x} \sqrt{x^2 + c^2} = \frac{x}{\sqrt{x^2 + c^2}}\).

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypotenuse.

Definition at line 46 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( double  x1, const fvar< T > &  x2  )

inline

Return the length of the hypotenuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypotenuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 66 of file hypot.hpp.

var stan::math::hypot ( double  a, const var &  b  )

inline

Returns the length of the hypotenuse of a right triangle with sides of the specified lengths (C99).

The derivative is

\(\frac{d}{d y} \sqrt{c^2 + y^2} = \frac{y}{\sqrt{c^2 + y^2}}\).

\[ \mbox{hypot}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \sqrt{x^2+y^2} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{x}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{y}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypotenuse.

Definition at line 91 of file hypot.hpp.

template<typename T >

std::complex<T> stan::math::i_times ( const std::complex< T > &  z )

inline

Return the specified complex number multiplied by i.

This compound function is more efficient than mulitplying by a constant i because it involves only a single arithmetic negation.

Template Parameters

value

type of complex argument

Parameters

[in]

z

complex argument

Returns

argument multipled by i

Definition at line 20 of file i_times.hpp.

template<bool Jacobian = false, typename T , typename... Types, require_all_not_var_matrix_t< T, Types... > * = nullptr>

auto stan::math::identity_constrain ( T &&  x, Types &&  ...  )

inline

Returns the result of applying the identity constraint transform to the input.

This promotes the input to the least upper bound of the input types.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	type of value to promote
Types	Other types.

Parameters

[in]

x

object

Returns

transformed input

This specialization handles convert Eigen matrices of doubles to var matrix types.

Template Parameters

T	Any type.
Types	Any type with one of T and Types being a var_value matrix.

Parameters

[in]

x

object

Returns

transformed input

Definition at line 23 of file identity_constrain.hpp.

template<typename T , typename... Types, require_all_not_var_matrix_t< T, Types... > * = nullptr>

auto stan::math::identity_free ( T &&  x, Types &&  ...  )

inline

Returns the result of applying the inverse of the identity constraint transform to the input.

Returns the result of applying the identity constraint transform to the input.

This promotes the input to the least upper bound of the input types.

Template Parameters

T	type of value to promote
Types	Other types.

Parameters

[in]

x

value to promote

Returns

value

For any of the inputs being a var_value type this promotes the input x to var_value.

Template Parameters

T	Any type.
Types	Any type with one of T and Types being a var_value matrix.

Parameters

[in]

x

object

Returns

transformed input

Definition at line 22 of file identity_free.hpp.

template<typename T_x , require_matrix_cl_t< T_x > * = nullptr>

auto stan::math::identity_matrix ( int  K )

inline

Return a square identity matrix.

Parameters

K	size of the matrix

Returns

An identity matrix of size K.

Exceptions

std::domain_error

if K is negative.

Definition at line 19 of file identity_matrix.hpp.

var stan::math::if_else ( bool  c, const var &  y_true, const var &  y_false  )

inline

If the specified condition is true, return the first variable, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 18 of file if_else.hpp.

template<typename T_true , typename T_false >

return_type_t<T_true, T_false> stan::math::if_else ( const bool  c, const T_true  y_true, const T_false  y_false  )

inline

Return the second argument if the first argument is true and otherwise return the second argument.

This is just a convenience method to provide a function with the same behavior as the built-in ternary operator. In general, this function behaves as if defined by

if_else(c, y1, y0) = c ? y1 : y0.

Template Parameters

T_true	type of the true argument
T_false	type of the false argument

Parameters

c	Boolean condition value.
y_true	Value to return if condition is true.
y_false	Value to return if condition is false.

Definition at line 26 of file if_else.hpp.

var stan::math::if_else ( bool  c, double  y_true, const var &  y_false  )

inline

If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Value to promote to variable and return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 30 of file if_else.hpp.

var stan::math::if_else ( bool  c, const var &  y_true, double  y_false  )

inline

If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Value to promote to variable and return if condition is false.

Definition at line 46 of file if_else.hpp.

template<typename T , require_autodiff_t< T > >

T stan::math::imag

(

const std::complex< T > &

z

)

Return the imaginary component of the complex argument.

Template Parameters

T	value type of complex argument

Parameters

[in]

z

complex value whose imaginary component is extracted

Returns

imaginary component of argument

Definition at line 18 of file imag.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, const fvar< T > &  b, const fvar< T > &  x  )

inline

Definition at line 19 of file inc_beta.hpp.

double stan::math::inc_beta ( double  a, double  b, double  x  )

inline

The normalized incomplete beta function of a, b, with outcome x.

Used to compute the cumulative density function for the beta distribution.

Parameters

a	Shape parameter a >= 0; a and b can't both be 0
b	Shape parameter b >= 0
x	Random variate. 0 <= x <= 1

Exceptions

if	constraints are violated or if any argument is NaN

Returns

The normalized incomplete beta function.

Definition at line 24 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, const fvar< T > &  b, const fvar< T > &  x  )

inline

Definition at line 32 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, double  b, const fvar< T > &  x  )

inline

Definition at line 36 of file inc_beta.hpp.

var stan::math::inc_beta ( const var &  a, const var &  b, const var &  c  )

inline

Definition at line 38 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, const fvar< T > &  b, double  x  )

inline

Definition at line 40 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, double  b, const fvar< T > &  x  )

inline

Definition at line 45 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, double  b, double  x  )

inline

Definition at line 49 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, const fvar< T > &  b, double  x  )

inline

Definition at line 53 of file inc_beta.hpp.

template<typename T >

T stan::math::inc_beta_dda	(	T	a,
		T	b,
		T	z,
		T	digamma_a,
		T	digamma_ab
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a.

The power series used to compute the derivative tends to converge slowly when a and b are large, especially if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters

T	scalar types of arguments

Parameters

a	first argument
b	second argument
z	upper bound of the integral
digamma_a	value of digamma(a)
digamma_ab	value of digamma(b)

Returns

partial derivative of the incomplete beta with respect to a

Precondition

a >= 0

b >= 0

0 <= z <= 1

Definition at line 39 of file inc_beta_dda.hpp.

template<typename T >

T stan::math::inc_beta_ddb	(	T	a,
		T	b,
		T	z,
		T	digamma_b,
		T	digamma_ab
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b.

The power series used to compute the derivative tends to converge slowly when a and b are large, especially if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters

T	scalar types of arguments

Parameters

a	first argument
b	second argument
z	upper bound of the integral
digamma_b	value of digamma(b)
digamma_ab	value of digamma(b)

Returns

partial derivative of the incomplete beta with respect to b

Precondition

a >= 0

b >= 0

0 <= z <= 1

Definition at line 42 of file inc_beta_ddb.hpp.

template<typename T >

T stan::math::inc_beta_ddz	(	T	a,
		T	b,
		T	z
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z.

Template Parameters

T	scalar types of arguments

Parameters

a	first argument
b	second argument
z	upper bound of the integral

Returns

partial derivative of the incomplete beta with respect to z

Precondition

a > 0

b > 0

0 < z <= 1

Definition at line 29 of file inc_beta_ddz.hpp.

template<>

double stan::math::inc_beta_ddz ( double  a, double  b, double  z  )

inline

Definition at line 37 of file inc_beta_ddz.hpp.

template<std::size_t N, class F >

constexpr auto stan::math::index_apply

(

F

f

)

Calls given callable with an index sequence.

Combination of this and a lambda can be used to avoid separating implementation that needs an index sequence from functions that operate on tuples of parameter packs.

Template Parameters

N	size of index sequence to use
F	callable type

Parameters

f

callable

Returns

what the given callable returns

Definition at line 26 of file index_apply.hpp.

void stan::math::indexing_rev	(	matrix_cl< double > &	adj,
		const matrix_cl< int > &	idx,
		const matrix_cl< double > &	res
	)

Performs reverse pass for indexing operation on the OpenCL device.

Depending on the size of indexed matrix and the amount of local memory available on the device selects the best kernel to use for the operation.

Parameters

[in,out]	adj	adjoint of the argument to indexing
	idx	indices
	res	adjoint of the result of the indexing operation

Definition at line 22 of file indexing_rev.hpp.

tbb::task_scheduler_init& stan::math::init_threadpool_tbb ( int  n_threads = 0 )

inline

Initialize the Intel TBB threadpool and global scheduler through the tbb::task_scheduler_init object.

In case an instance of the tbb::task_scheduler_object has been instantiated prior to calling this function, then any subsequent initialization is ignored by the Intel TBB.

The maximal number of threads is read from the environment variable STAN_NUM_THREADS using internal::get_num_threads. See conventions of get_num_threads. The TBB scheduler will be activated by calling this function.

The function returns a reference to the static tbb::task_scheduler_init instance.

Parameters

n_threads

The maximum number of threads available to the tbb. If not set will search for the environment variable STAN_NUM_THREADS. A value of will assume all detectable threads are available for the process.

Returns

reference to the static tbb::task_scheduler_init

Exceptions

std::runtime_error

if n_threads (defaults to zero) is not provided and the value of STAN_NUM_THREADS environment variable is invalid.

Definition at line 145 of file init_threadpool_tbb.hpp.

template<typename T , typename V , require_all_stan_scalar_t< T, V > * = nullptr, require_convertible_t< V, T > * = nullptr>

void stan::math::initialize ( T &  x, V  v  )

inline

Definition at line 17 of file initialize.hpp.

template<typename T , int R, int C, typename V >

void stan::math::initialize ( Eigen::Matrix< T, R, C > &  x, const V &  v  )

inline

Definition at line 22 of file initialize.hpp.

template<typename T , typename V >

void stan::math::initialize ( std::vector< T > &  x, const V &  v  )

inline

Definition at line 27 of file initialize.hpp.

template<typename EigMat , typename S , require_eigen_t< EigMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>

void stan::math::initialize_fill ( EigMat &  x, const S &  y  )

inline

Fill the specified container with the specified value.

The specified matrix is filled by element.

Template Parameters

EigMat	Type inheriting from EigenBase
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 24 of file initialize_fill.hpp.

template<typename VarMat , typename S , require_var_matrix_t< VarMat > * = nullptr, require_stan_scalar_t< S > * = nullptr>

void stan::math::initialize_fill ( VarMat &  x, const S &  y  )

inline

Fill the specified container with the specified value.

This function does not perform a callback to propogate the adjoints upward

The specified matrix is filled by element.

Template Parameters

VarMat	a var_value with inner type from EigenBase
S	A var.

Parameters

x	Container.
y	Value.

Definition at line 26 of file initialize_fill.hpp.

template<typename T , typename S , require_t< std::is_assignable< std::decay_t< T > &, std::decay_t< S >>> * = nullptr>

void stan::math::initialize_fill ( T &  x, S &&  y  )

inline

Fill the specified container with the specified value.

This base case simply assigns the value to the container.

Template Parameters

T	Type of reference container.
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 41 of file initialize_fill.hpp.

template<typename Vec , typename S , require_std_vector_t< Vec > * = nullptr>

void stan::math::initialize_fill ( Vec &  x, S &&  y  )

inline

Fill the specified container with the specified value.

Each container in the specified standard vector is filled recursively by calling fill.

Template Parameters

Vec	A standard vector
S	type of value

Parameters

[in]	x	Container.
[in,out]	y	Value.

Definition at line 57 of file initialize_fill.hpp.

void stan::math::initialize_variable ( var &  variable, const var &  value  )

inline

Initialize variable to value.

(Function may look pointless, but it's needed to bottom out recursion.)

Definition at line 16 of file initialize_variable.hpp.

template<int R, int C>

void stan::math::initialize_variable ( Eigen::Matrix< var, R, C > &  matrix, const var &  value  )

inline

Initialize every cell in the matrix to the specified value.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Definition at line 27 of file initialize_variable.hpp.

template<typename T >

void stan::math::initialize_variable ( std::vector< T > &  variables, const var &  value  )

inline

Initialize the variables in the standard vector recursively.

Template Parameters

T	type of elements in the vector

Definition at line 38 of file initialize_variable.hpp.

template<typename T >

int stan::math::int_step ( const T &  y )

inline

The integer step, or Heaviside, function.

For double NaN input, int_step(NaN) returns 0.

\[ \mbox{int\_step}(x) = \begin{cases} 0 & \mbox{if } x \leq 0 \\ 1 & \mbox{if } x > 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	value type

Parameters

[in]

y

value

Returns

1 if value is greater than 0 and 0 otherwise

Definition at line 27 of file int_step.hpp.

template<typename F >

double stan::math::integrate ( const F &  f, double  a, double  b, double  relative_tolerance  )

inline

Integrate a single variable function f from a to b to within a specified relative tolerance.

This function assumes a is less than b.

The signature for f should be: double f(double x, double xc)

It should return the value of the function evaluated at x.

Depending on whether or not a is finite or negative infinity and b is finite or positive infinity, a different version of the 1d quadrature algorithm from the Boost quadrature library is chosen.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

Template Parameters

T

Type of f

Parameters

f	the function to be integrated
a	lower limit of integration
b	upper limit of integration
relative_tolerance	target relative tolerance passed to Boost quadrature

Returns

numeric integral of function f

If the integral crosses zero, break it into two (advice from the Boost implementation: https://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/double_exponential/de_caveats.html)

Definition at line 52 of file integrate_1d.hpp.

template<typename F , typename T_a , typename T_b , typename T_theta , typename = require_any_var_t<T_a, T_b, T_theta>>

return_type_t<T_a, T_b, T_theta> stan::math::integrate_1d ( const F &  f, const T_a &  a, const T_b &  b, const std::vector< T_theta > &  theta, const std::vector< double > &  x_r, const std::vector< int > &  x_i, std::ostream *  msgs, const double  relative_tolerance = std::sqrt(EPSILON)  )

inline

Compute the integral of the single variable function f from a to b to within a specified relative tolerance.

a and b can be finite or infinite.

f should be compatible with reverse mode autodiff and have the signature: var f(double x, double xc, const std::vector<var>& theta, const std::vector<double>& x_r, const std::vector<int> &x_i, std::ostream* msgs)

It should return the value of the function evaluated at x. Any errors should be printed to the msgs stream.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

The integration algorithm terminates when

\[ \frac{{|I_{n + 1} - I_n|}}{{|I|_{n + 1}}} < \text{relative tolerance} \]

where \(I_{n}\) is the nth estimate of the integral and \(|I|_{n}\) is the nth estimate of the norm of the integral.

Integrals that cross zero are split into two. In this case, each integral is separately integrated to the given relative_tolerance.

Gradients of f that evaluate to NaN when the function evaluates to zero are set to zero themselves. This is due to the autodiff easily overflowing to NaN when evaluating gradients near the maximum and minimum floating point values (where the function should be zero anyway for the integral to exist)

Template Parameters

T_a	type of first limit
T_b	type of second limit
T_theta	type of parameters
T	Type of f

Parameters

	f	the functor to integrate
	a	lower limit of integration
	b	upper limit of integration
	theta	additional parameters to be passed to f
	x_r	additional data to be passed to f
	x_i	additional integer data to be passed to f
[in,out]	msgs	the print stream for warning messages
	relative_tolerance	relative tolerance passed to Boost quadrature

Returns

numeric integral of function f

Definition at line 215 of file integrate_1d.hpp.

template<typename F >

double stan::math::integrate_1d ( const F &  f, double  a, double  b, const std::vector< double > &  theta, const std::vector< double > &  x_r, const std::vector< int > &  x_i, std::ostream *  msgs, const double  relative_tolerance = std::sqrt(EPSILON)  )

inline

Compute the integral of the single variable function f from a to b to within a specified relative tolerance.

a and b can be finite or infinite.

The signature for f should be: double f(double x, double xc, const std::vector<double>& theta, const std::vector<double>& x_r, const std::vector<int>& x_i, std::ostream* msgs)

It should return the value of the function evaluated at x. Any errors should be printed to the msgs stream.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

The integration algorithm terminates when

\[ \frac{{|I_{n + 1} - I_n|}}{{|I|_{n + 1}}} < \text{relative tolerance} \]

where \(I_{n}\) is the nth estimate of the integral and \(|I|_{n}\) is the nth estimate of the norm of the integral.

Integrals that cross zero are split into two. In this case, each integral is separately integrated to the given relative_tolerance.

Template Parameters

T

Type of f

Parameters

	f	the function to be integrated
	a	lower limit of integration
	b	upper limit of integration
	theta	additional parameters to be passed to f
	x_r	additional data to be passed to f
	x_i	additional integer data to be passed to f
[in,out]	msgs	the print stream for warning messages
	relative_tolerance	tolerance passed to Boost quadrature

Returns

numeric integral of function f

Definition at line 240 of file integrate_1d.hpp.

template<typename F , typename T_a , typename T_b , typename... Args, require_any_st_var< T_a, T_b, Args... > * = nullptr>

return_type_t<T_a, T_b, Args...> stan::math::integrate_1d_impl ( const F &  f, const T_a &  a, const T_b &  b, double  relative_tolerance, std::ostream *  msgs, const Args &...  args  )

inline

Return the integral of f from a to b to the given relative tolerance.

Template Parameters

F	Type of f
T_a	type of first limit
T_b	type of second limit
Args	types of parameter pack arguments

Parameters

	f	the functor to integrate
	a	lower limit of integration
	b	upper limit of integration
	relative_tolerance	relative tolerance passed to Boost quadrature
[in,out]	msgs	the print stream for warning messages
	args	additional arguments to pass to f

Returns

numeric integral of function f

Definition at line 40 of file integrate_1d.hpp.

template<typename F , typename... Args, require_all_not_st_var< Args... > * = nullptr>

double stan::math::integrate_1d_impl ( const F &  f, double  a, double  b, double  relative_tolerance, std::ostream *  msgs, const Args &...  args  )

inline

Compute the integral of the single variable function f from a to b to within a specified relative tolerance.

a and b can be finite or infinite.

Template Parameters

T

Type of f

Parameters

	f	the function to be integrated
	a	lower limit of integration
	b	upper limit of integration
	relative_tolerance	tolerance passed to Boost quadrature
[in,out]	msgs	the print stream for warning messages
	args	additional arguments passed to f

Returns

numeric integral of function f

Definition at line 176 of file integrate_1d.hpp.

template<typename F , typename Tpar >

std::vector<std::vector<Tpar> > stan::math::integrate_dae	(	const F &	f,
		const std::vector< double > &	yy0,
		const std::vector< double > &	yp0,
		double	t0,
		const std::vector< double > &	ts,
		const std::vector< Tpar > &	theta,
		const std::vector< double > &	x_r,
		const std::vector< int > &	x_i,
		const double	rtol,
		const double	atol,
		const int64_t	max_num_steps = idas_integrator::IDAS_MAX_STEPS,
		std::ostream *	msgs = nullptr
	)

Return the solutions for a semi-explicit DAE system with residual specified by functor F, given the specified consistent initial state yy0 and yp0.

Template Parameters

DAE	type of DAE system
Tpar	scalar type of parameter theta

Parameters

[in]	f	functor for the base ordinary differential equation
[in]	yy0	initial state
[in]	yp0	initial derivative state
[in]	t0	initial time
[in]	ts	times of the desired solutions, in strictly increasing order, all greater than the initial time
[in]	theta	parameters
[in]	x_r	real data
[in]	x_i	int data
[in]	rtol	relative tolerance passed to IDAS, required <10^-3
[in]	atol	absolute tolerance passed to IDAS, problem-dependent
[in]	max_num_steps	maximal number of admissable steps between time-points
[in]	msgs	message

Returns

a vector of states, each state being a vector of the same size as the state variable, corresponding to a time in ts.

Definition at line 39 of file integrate_dae.hpp.

template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >

std::vector<std::vector<return_type_t<T_y0, T_param, T_t0, T_ts> > > stan::math::integrate_ode_adams	(	const F &	f,
		const std::vector< T_y0 > &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		const std::vector< T_param > &	theta,
		const std::vector< double > &	x,
		const std::vector< int > &	x_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	absolute_tolerance = 1e-10,
		long int	max_num_steps = 1e8
	)

Deprecated:

use ode_adams

Definition at line 19 of file integrate_ode_adams.hpp.

template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >

std::vector<std::vector<return_type_t<T_y0, T_param, T_t0, T_ts> > > stan::math::integrate_ode_bdf	(	const F &	f,
		const std::vector< T_y0 > &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		const std::vector< T_param > &	theta,
		const std::vector< double > &	x,
		const std::vector< int > &	x_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	absolute_tolerance = 1e-10,
		long int	max_num_steps = 1e8
	)

Deprecated:

use ode_bdf

Definition at line 19 of file integrate_ode_bdf.hpp.

template<typename F , typename T_y0 , typename T_param , typename T_t0 , typename T_ts >

auto stan::math::integrate_ode_rk45 ( const F &  f, const std::vector< T_y0 > &  y0, const T_t0 &  t0, const std::vector< T_ts > &  ts, const std::vector< T_param > &  theta, const std::vector< double > &  x, const std::vector< int > &  x_int, std::ostream *  msgs = nullptr, double  relative_tolerance = 1e-6, double  absolute_tolerance = 1e-6, int  max_num_steps = 1e6  )

inline

Deprecated:

use ode_rk45

Definition at line 18 of file integrate_ode_rk45.hpp.

template<typename T >

fvar<T> stan::math::inv ( const fvar< T > &  x )

inline

Definition at line 12 of file inv.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv ( const var_value< T > &  A )

inline

Returns the elementwise inv() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv() of the input.

Definition at line 20 of file inv.hpp.

template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>

auto stan::math::inv ( T_x &&  x )

inline

Return the elementwise 1.0 / x of the specified argument, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x	Input kernel generator expression.

Returns

elementwise 1.0 / x of the specified argument.

Definition at line 21 of file inv.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::inv ( const var_value< T > &  a )

inline

Template Parameters

T	Arithmetic or a type inheriting from EigenBase. \[ \mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv}(x)}{\partial x} = \begin{cases} -\frac{1}{x^2} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 31 of file inv.hpp.

template<typename T , require_not_container_st< std::is_arithmetic, T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::inv ( const T &  x )

inline

Return the elementwise 1.0 / x of the specified argument, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

1 divided by each value in x.

Definition at line 37 of file inv.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::inv ( const Container &  x )

inline

Version of inv() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

1 divided by each value in x.

Definition at line 53 of file inv.hpp.

template<typename T_y , typename T_dof >

return_type_t<T_y, T_dof> stan::math::inv_chi_square_lpdf ( const T_y &  y, const T_dof &  nu  )

inline

Definition at line 103 of file inv_chi_square_lpdf.hpp.

template<typename T >

fvar<T> stan::math::inv_cloglog ( const fvar< T > &  x )

inline

Definition at line 13 of file inv_cloglog.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv_cloglog ( const var_value< T > &  A )

inline

Returns the elementwise inv_cloglog() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv_cloglog() of the input.

Definition at line 20 of file inv_cloglog.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::inv_cloglog ( const var_value< T > &  a )

inline

Return the inverse complementary log-log function applied specified variable (stan).

See inv_cloglog() for the double-based version.

The derivative is given by

\(\frac{d}{dx} \mbox{cloglog}^{-1}(x) = \exp (x - \exp (x))\).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Variable argument.

Returns

The inverse complementary log-log of the specified argument.

Definition at line 28 of file inv_cloglog.hpp.

double stan::math::inv_cloglog ( double  x )

inline

The inverse complementary log-log function.

The function is defined by

inv_cloglog(x) = 1 - exp(-exp(x)).

This function can be used to implement the inverse link function for complementary-log-log regression.

\[ \mbox{inv\_cloglog}(y) = \begin{cases} \mbox{cloglog}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_cloglog}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{cloglog}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{cloglog}^{-1}(y) = 1 - \exp \left( - \exp(y) \right) \]

\[ \frac{\partial \, \mbox{cloglog}^{-1}(y)}{\partial y} = \exp(y-\exp(y)) \]

Parameters

x

Argument.

Returns

Inverse complementary log-log of the argument.

Definition at line 48 of file inv_cloglog.hpp.

template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>

auto stan::math::inv_cloglog ( T_x &&  x )

inline

The inverse complementary log-log function.

The function is defined by

inv_cloglog(x) = 1 - exp(-exp(x)).

This function can be used to implement the inverse link function for complementary-log-log regression.

\[ \mbox{inv\_cloglog}(y) = \begin{cases} \mbox{cloglog}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_cloglog}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{cloglog}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{cloglog}^{-1}(y) = 1 - \exp \left( - \exp(y) \right) \]

\[ \frac{\partial \, \mbox{cloglog}^{-1}(y)}{\partial y} = \exp(y-\exp(y)) \]

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x

Argument.

Returns

Inverse complementary log-log of the argument.

Definition at line 53 of file inv_cloglog.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::inv_cloglog ( const Container &  x )

inline

Vectorized version of inv_cloglog().

Template Parameters

Container

type of container

Parameters

x

container

Returns

1 - exp(-exp()) applied to each value in x.

Definition at line 79 of file inv_cloglog.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::inv_cloglog ( const Container &  x )

inline

Version of inv_cloglog() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

1 - exp(-exp()) applied to each value in x.

Definition at line 93 of file inv_cloglog.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	beta
	)

Definition at line 25 of file inv_gamma_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	beta
	)

Definition at line 25 of file inv_gamma_lcdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  beta  )

inline

Definition at line 116 of file inv_gamma_lpdf.hpp.

template<typename T >

fvar<T> stan::math::inv_logit ( const fvar< T > &  x )

inline

Returns the inverse logit function applied to the argument.

Template Parameters

T	inner type of the fvar

Parameters

x

argument

Returns

inverse logit of argument

Definition at line 20 of file inv_logit.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv_logit ( const var_value< T > &  A )

inline

Returns the elementwise inv_logit() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv_logit() of the input.

Definition at line 20 of file inv_logit.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::inv_logit ( const var_value< T > &  a )

inline

The inverse logit function for variables (stan).

See inv_logit() for the double-based version.

The derivative of inverse logit is

\(\frac{d}{dx} \mbox{logit}^{-1}(x) = \mbox{logit}^{-1}(x) (1 - \mbox{logit}^{-1}(x))\).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Argument variable.

Returns

Inverse logit of argument.

Definition at line 26 of file inv_logit.hpp.

double stan::math::inv_logit ( double  a )

inline

Returns the inverse logit function applied to the argument.

The inverse logit function is defined by

\(\mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(-x)}\).

This function can be used to implement the inverse link function for logistic regression.

The inverse to this function is logit.

\[ \mbox{inv\_logit}(y) = \begin{cases} \mbox{logit}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_logit}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{logit}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{logit}^{-1}(y) = \frac{1}{1 + \exp(-y)} \]

\[ \frac{\partial \, \mbox{logit}^{-1}(y)}{\partial y} = \frac{\exp(y)}{(\exp(y)+1)^2} \]

Parameters

a

Argument.

Returns

Inverse logit of argument.

Definition at line 51 of file inv_logit.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::inv_logit ( const T &  x )

inline

Vectorized version of inv_logit().

Template Parameters

T	type of container

Parameters

x

container

Returns

Inverse logit applied to each value in x.

Definition at line 87 of file inv_logit.hpp.

template<typename T >

fvar<T> stan::math::inv_Phi ( const fvar< T > &  p )

inline

Definition at line 15 of file inv_Phi.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv_Phi ( const var_value< T > &  A )

inline

Returns the elementwise inv_Phi() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv_Phi() of the input.

Definition at line 20 of file inv_Phi.hpp.

var stan::math::inv_Phi ( const var &  p )

inline

The inverse of unit normal cumulative density function.

The derivative is the reciprocal of unit normal density function,

Parameters

p	Probability

Returns

The unit normal inverse cdf evaluated at p

Definition at line 21 of file inv_Phi.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::inv_Phi ( const T &  p )

inline

Return the elementwise inverse of unit normal cumulative density function.

Template Parameters

T	a var_value with inner Eigen type

Parameters

p	Probability vector

Returns

Elementwise unit normal inverse cdf

Definition at line 35 of file inv_Phi.hpp.

double stan::math::inv_Phi ( double  p )

inline

Return the value of the inverse standard normal cumulative distribution function at the specified argument.

The precision is at or better than 1.5e-15 for values between 0.0000001 he largest integer that protects against floating point errors for the inv_Phi function. The value was found by finding the largest integer that passed the unit tests for accuracy when the input into inv_Phi is near 1.

Parameters

p	argument between 0 and 1 inclusive

Returns

real value of the inverse cdf for the standard normal distribution

Definition at line 147 of file inv_Phi.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::inv_Phi ( const T &  x )

inline

Vectorized version of inv_Phi().

Template Parameters

T	type of container

Parameters

x	variables in range [0, 1]

Returns

Inverse unit normal CDF of each value in x.

Exceptions

std::domain_error

if any value is not between 0 and 1.

Definition at line 181 of file inv_Phi.hpp.

template<typename T >

fvar<T> stan::math::inv_sqrt ( const fvar< T > &  x )

inline

Definition at line 12 of file inv_sqrt.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

auto stan::math::inv_sqrt ( T  x )

inline

Definition at line 16 of file inv_sqrt.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv_sqrt ( const var_value< T > &  A )

inline

Returns the elementwise inv_sqrt() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv_sqrt() of the input.

Definition at line 20 of file inv_sqrt.hpp.

template<typename T_x , typename = require_all_kernel_expressions_and_none_scalar_t<T_x>>

auto stan::math::inv_sqrt ( T_x &&  x )

inline

Return the elementwise 1 / sqrt(x) of the specified kernel generator expression.

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x	input kernel generator expression

Returns

inverse square root of each value in x.

Definition at line 21 of file inv_sqrt.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::inv_sqrt ( const var_value< T > &  a )

inline

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

\[ \mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_sqrt}(x)}{\partial x} = \begin{cases} -\frac{1}{2\sqrt{x^3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 34 of file inv_sqrt.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_stan_scalar_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::inv_sqrt ( const Container &  x )

inline

Return the elementwise 1 / sqrt(x)} of the specified argument, which may be a scalar or any Stan container of numeric scalars.

Template Parameters

Container

type of container

Parameters

x

container

Returns

inverse square root of each value in x.

Definition at line 48 of file inv_sqrt.hpp.

template<typename Container , require_not_var_matrix_t< Container > * = nullptr, require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::inv_sqrt ( const Container &  x )

inline

Version of inv_sqrt() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

inverse square root each variable in the container.

Definition at line 62 of file inv_sqrt.hpp.

template<typename T >

fvar<T> stan::math::inv_square ( const fvar< T > &  x )

inline

Definition at line 12 of file inv_square.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::inv_square ( const var_value< T > &  A )

inline

Returns the elementwise inv_square() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise inv_square() of the input.

Definition at line 20 of file inv_square.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::inv_square ( const Container &  x )

inline

Returns 1 / square(x).

Template Parameters

Container

type of container

Parameters

x

container

Returns

1 / square(x) of each value in x.

Definition at line 23 of file inv_square.hpp.

var stan::math::inv_square ( const var &  a )

inline

\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_square}(x)}{\partial x} = \begin{cases} -\frac{2}{x^3} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 30 of file inv_square.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::inv_square ( const Container &  x )

inline

Version of inv_square() that accepts Eigen Matrix/Array objects or expressions.

Template Parameters

T

Type of x

Parameters

x	Eigen Matrix/Array or expression

Returns

1 / the square of each value in x.

Definition at line 37 of file inv_square.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::inv_wishart_lpdf ( const T_y &  W, const T_dof &  nu, const T_scale &  S  )

inline

Definition at line 92 of file inv_wishart_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::inv_wishart_rng ( double  nu, const Eigen::MatrixXd &  S, RNG &  rng  )

inline

Definition at line 13 of file inv_wishart_rng.hpp.

template<typename T >

void stan::math::invalid_argument ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::invalid_argument

Definition at line 27 of file invalid_argument.hpp.

template<typename T >

void stan::math::invalid_argument ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once. (We've already changed behavior multiple times up to Stan v2.5.0.) The message is: "<function>: <name> <msg1><y>"

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
msg1	Message to print before the variable

Exceptions

std::invalid_argument

Definition at line 49 of file invalid_argument.hpp.

template<typename T >

void stan::math::invalid_argument_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior multiple times up to Stan v2.5.0.) The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::invalid_argument

Definition at line 33 of file invalid_argument_vec.hpp.

template<typename T >

void stan::math::invalid_argument_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg  )

inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior multiple times up to Stan v2.5.0.) The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg	Message to print before the variable

Exceptions

std::invalid_argument

Definition at line 61 of file invalid_argument_vec.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::inverse ( const T &  m )

inline

Reverse mode specialization of calculating the inverse of the matrix.

Parameters

m	specified matrix

Returns

Inverse of the matrix (an empty matrix if the specified matrix has size zero).

Exceptions

std::invalid_argument

if the matrix is not square.

Definition at line 23 of file inverse.hpp.

template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>

Eigen::Matrix< value_type_t< EigMat >, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime > stan::math::inverse ( const EigMat &  m )

inline

Forward mode specialization of calculating the inverse of the matrix.

Returns the inverse of the specified matrix.

Template Parameters

T	type of elements in the matrix

Parameters

m	specified matrix

Returns

Inverse of the matrix (an empty matrix if the specified matrix has size zero).

Exceptions

std::invalid_argument

if the matrix is not square.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	specified matrix

Returns

Inverse of the matrix (an empty matrix if the specified matrix has size zero).

Exceptions

std::invalid_argument

if the matrix is not square.

Definition at line 29 of file inverse.hpp.

template<typename Vector , require_vector_t< Vector > * = nullptr>

void stan::math::inverse_softmax	(	const Vector &	simplex,
		Vector &	y
	)

Writes the inverse softmax of the simplex argument into the second argument.

See softmax for the inverse function and a definition of the relation.

The inverse softmax function is defined by

\(\mbox{inverse\_softmax}(x)[i] = \log x[i]\).

This function defines the inverse of softmax up to a scaling factor.

Because of the definition, values of 0.0 in the simplex are converted to negative infinity, and values of 1.0 are converted to 0.0.

There is no check that the input vector is a valid simplex vector.

Template Parameters

Vector

type of the simplex vector

Parameters

simplex	Simplex vector input.
y	Vector into which the inverse softmax is written.

Exceptions

std::invalid_argument

if size of the input and output vectors differ.

Definition at line 37 of file inverse_softmax.hpp.

template<typename EigMat >

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::inverse_spd ( const EigMat &  m )

inline

Returns the inverse of the specified symmetric, pos/neg-definite matrix.

Template Parameters

EigMat

type of elements in the matrix

Parameters

m	specified matrix

Returns

Inverse of the matrix (an empty matrix if the specified matrix has size zero).

Exceptions

std::invalid_argument	if the matrix is not symmetric.
std::domain_error	if the matrix is not positive definite.

Definition at line 23 of file inverse_spd.hpp.

template<typename T >

bool stan::math::is_aligned	(	T *	ptr,
		unsigned int	bytes_aligned
	)

Return true if the specified pointer is aligned on the number of bytes.

This doesn't really make sense other than for powers of 2.

Parameters

ptr	Pointer to test.
bytes_aligned	Number of bytes of alignment required.

Returns

true if pointer is aligned.

Template Parameters

Type	of object to which pointer points.

Definition at line 29 of file stack_alloc.hpp.

template<typename T >

bool stan::math::is_any_nan ( const T &  x )

inline

Returns true if the input is NaN and false otherwise.

Delegates to stan::math::is_nan so that appropriate specializations can be loaded for autodiff types.

Parameters

x	Value to test.

Returns

true if the value is NaN.

Definition at line 21 of file is_any_nan.hpp.

template<typename T , typename... Ts>

bool stan::math::is_any_nan ( const T &  x, const Ts &...  xs  )

inline

Returns true if any input is NaN and false otherwise.

Delegates to stan::math::is_nan.

Parameters

x	first argument
xs	parameter pack of remaining arguments to forward to function

Returns

true if any value is NaN

Definition at line 35 of file is_any_nan.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_cholesky_factor ( const EigMat &  y )

inline

Return true if y is a valid Cholesky factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true if y is a valid Cholesky factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN

Definition at line 29 of file is_cholesky_factor.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_cholesky_factor_corr ( const EigMat &  y )

inline

Return true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no zero columns, and no element in matrix is NaN.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. This definition does not require a square matrix.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN

Definition at line 26 of file is_cholesky_factor_corr.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

bool stan::math::is_column_index ( const EigMat &  y, size_t  i  )

inline

Return true if column index is in bounds.

By default this is a 1-indexed check (as opposed to zero-indexed). Behavior can be changed by setting stan::error_index::value.

Template Parameters

EigMat

A type derived from EigenBase

Parameters

y	matrix to test
i	column index to check

Returns

true if column index is in bounds

Definition at line 20 of file is_column_index.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_corr_matrix ( const EigMat &  y )

inline

Return true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive).

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN

Definition at line 30 of file is_corr_matrix.hpp.

bool stan::math::is_inf ( double  x )

inline

Returns true if the input is infinite and false otherwise.

Delegates to std::isinf.

Parameters

x	Value to test.

Returns

true if the value is infinite.

Definition at line 18 of file is_inf.hpp.

int stan::math::is_inf ( const var &  v )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to is_inf.

Parameters

v	Value to test.

Returns

1 if the value is infinite and 0 otherwise.

Definition at line 20 of file is_inf.hpp.

template<typename T >

int stan::math::is_inf ( const fvar< T > &  x )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to is_inf.

Template Parameters

T	inner type of the fvar

Parameters

x	Value to test.

Returns

1 if the value is infinite and 0 otherwise.

Definition at line 21 of file is_inf.hpp.

template<typename T >

bool stan::math::is_integer ( T  x )

inline

Returns true if the input is an integer and false otherwise.

Parameters

x	Value to test.

Returns

true if the value is an integer

Definition at line 18 of file is_integer.hpp.

template<typename T >

bool stan::math::is_ldlt_factor ( LDLT_factor< T > &  A )

inline

Return true if the specified LDLT factor is invalid.

An LDLT_factor is invalid if it was constructed from a matrix that is not positive definite.

Template Parameters

T	Type matrix of LDLT

Parameters

A	The LDLT factor to check for validity

Returns

true if the LDLT factor is valid

Definition at line 21 of file is_ldlt_factor.hpp.

template<typename T_y , typename T_high >

bool stan::math::is_less_or_equal ( const T_y &  y, const T_high &  high  )

inline

Return true if y is less or equal to high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type of y
T_high	Type of upper bound

Parameters

y	Variable to check
high	Upper bound

Returns

true if y is less than or equal to low and if and element of y or high is NaN

Definition at line 26 of file is_less_or_equal.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_lower_triangular ( const EigMat &  y )

inline

Return true is matrix is lower triangular.

A matrix x is not lower triangular if there is a non-zero entry x[m, n] with m < n. This function only inspect the upper and triangular portion of the matrix, not including the diagonal.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true is matrix is lower triangular

Definition at line 21 of file is_lower_triangular.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

bool stan::math::is_mat_finite ( const EigMat &  y )

inline

Return true is the specified matrix is finite.

Template Parameters

T	Scalar type of the matrix, requires class method .allFinite()
EigMat	A type derived from EigenBase

Parameters

y	Matrix to test

Returns

true if the matrix is finite

Definition at line 20 of file is_mat_finite.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_matrix_t< EigMat1, EigMat2 > * = nullptr>

bool stan::math::is_matching_dims ( const EigMat1 &  y1, const EigMat2 &  y2  )

inline

Return true if the two matrices are of the same size.

This function checks the runtime sizes only.

Template Parameters

EigMat1	A type derived from EigenBase
EigMat2	A type derived from EigenBase

Parameters

y1	first matrix to test
y2	second matrix to test

Returns

true if the dimensions of the matrices match

Definition at line 22 of file is_matching_dims.hpp.

template<bool check_compile, typename EigMat1 , typename EigMat2 , require_all_matrix_t< EigMat1, EigMat2 > * = nullptr>

bool stan::math::is_matching_dims ( const EigMat1 &  y1, const EigMat2 &  y2  )

inline

Return true if the two matrices are of the same size.

This function checks the runtime sizes and can also check the static sizes as well. For example, a 4x1 matrix is not the same as a vector with 4 elements.

Template Parameters

check_compile	Whether to check the static sizes
EigMat1	A type derived from EigenBase
EigMat2	A type derived from EigenBase

Parameters

y1	first matrix to test
y2	second matrix to test

Returns

true if the dimensions of the matrices match

Definition at line 41 of file is_matching_dims.hpp.

template<typename T_y1 , typename T_y2 >

bool stan::math::is_matching_size ( const T_y1 &  y1, const T_y2 &  y2  )

inline

Return true if two structures are the same size.

This is a top-level sizing function for std::vector where a 2x3 vector returns 2, and a total sizing function for Eigen::Matrix where a 2x3 matrix will return 6.

Template Parameters

T_y1	Type of the first variable, requires class method .size()
T_y2	Type of the second variable, requires class method .size()

Parameters

y1	First variable
y2	Second variable

Returns

true if the variable sizes match

Definition at line 24 of file is_matching_size.hpp.

template<typename T , typename = require_arithmetic_t<T>>

bool stan::math::is_nan ( T  x )

inline

Returns true if the input is NaN and false otherwise.

Delegates to std::isnan.

Parameters

x	Value to test.

Returns

true if the value is NaN.

Definition at line 19 of file is_nan.hpp.

bool stan::math::is_nan ( const var &  v )

inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to is_nan(double).

Template Parameters

T	type of input

Parameters

v	value to test

Returns

1 if the value is NaN and 0 otherwise.

Definition at line 20 of file is_nan.hpp.

template<typename T , require_fvar_t< T > * = nullptr>

bool stan::math::is_nan ( T &&  x )

inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to is_nan.

Template Parameters

T	inner type of the fvar

Parameters

x	Value to test.

Returns

1 if the value is NaN and 0 otherwise.

Definition at line 22 of file is_nan.hpp.

template<typename T , typename = require_eigen_t<T>>

bool stan::math::is_nan ( const T &  x )

inline

Definition at line 24 of file is_nan.hpp.

template<typename T >

bool stan::math::is_nonpositive_integer ( T  x )

inline

Returns true if the input is a nonpositive integer and false otherwise.

Parameters

x	Value to test.

Returns

true if the value is an integer

Definition at line 18 of file is_nonpositive_integer.hpp.

template<typename T_y >

bool stan::math::is_nonzero_size ( const T_y &  y )

inline

Returns true if the specified matrix/vector is size nonzero.

Template Parameters

T_y	Type of container, requires class method .size()

Parameters

y	Container to test – matrix/vector

Returns

true if container has size zero

Definition at line 15 of file is_nonzero_size.hpp.

template<typename T_y >

bool stan::math::is_not_nan ( const T_y &  y )

inline

Return true if y is not NaN.

This function is vectorized and will check each element of y. If no element is NaN, this function will return true.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Returns

true if no element of y is NaN

Definition at line 23 of file is_not_nan.hpp.

template<typename T_y >

bool stan::math::is_ordered ( const std::vector< T_y > &  y )

inline

Return true if the vector is sorted into strictly increasing order.

Template Parameters

T_y	Type of scalar, requires class method .size()

Parameters

y	std::vector to test

Returns

true if vector is sorted in ascending order

Definition at line 19 of file is_ordered.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_pos_definite ( const EigMat &  y )

inline

Return true if the matrix is square or if the matrix has non-zero size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true if the matrix is square or if the matrix has non-0 size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN

Definition at line 27 of file is_pos_definite.hpp.

template<typename Derived >

bool stan::math::is_pos_definite ( const Eigen::LDLT< Derived > &  cholesky )

inline

Return true if the matrix is positive definite.

Template Parameters

Derived

Derived type of the Eigen::LDLT transform, requires class method .info() and .isPositive() and .vectorD().array()

Parameters

cholesky

Eigen::LDLT to test, whose progenitor must not have any NaN elements

Returns

true if the matrix is positive definite

Definition at line 56 of file is_pos_definite.hpp.

template<typename Derived >

bool stan::math::is_pos_definite ( const Eigen::LLT< Derived > &  cholesky )

inline

Return true if diagonal of the L matrix is positive.

Template Parameters

Derived

Derived type of the Eigen::LLT transform, requires class method .info() and .matrixLLT().diagonal().array()

Parameters

cholesky

Eigen::LLT to test, whose progenitor must not have any NaN elements

Returns

true if diagonal of the L matrix is positive

Definition at line 71 of file is_pos_definite.hpp.

template<typename T_y >

bool stan::math::is_positive ( const T_y &  y )

inline

Return true if y is positive.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Returns

true if vector contains only positive elements

Definition at line 20 of file is_positive.hpp.

bool stan::math::is_positive ( int  size )

inline

Return true if size is positive.

Parameters

size	Size value to check

Returns

true if size is not zero or negative

Definition at line 34 of file is_positive.hpp.

template<typename T_y >

bool stan::math::is_scal_finite ( const T_y &  y )

inline

Return true if y is finite.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Exceptions

<code>true</code>

if y is not infinity, -infinity, or NaN

Definition at line 22 of file is_scal_finite.hpp.

template<typename T_size1 , typename T_size2 >

bool stan::math::is_size_match ( T_size1  i, T_size2  j  )

inline

Return true if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

i	Size of variable 1
j	Size of variable 2

Returns

true if provided dimensions match

Definition at line 17 of file is_size_match.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

bool stan::math::is_square ( const EigMat &  y )

inline

Return true if the matrix is square.

This check allows 0x0 matrices.

Template Parameters

EigMat

A type derived from EigenBase

Parameters

y	Matrix to test

Returns

true if matrix is square

Definition at line 19 of file is_square.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr>

bool stan::math::is_symmetric ( const EigMat &  y )

inline

Return true if the matrix is square, and no element not on the main diagonal is NaN.

Template Parameters

EigMat

A type derived from EigenBase with dynamic rows and columns

Parameters

y	Matrix to test

Returns

true if the matrix is square, and no element not on the main diagonal is NaN

Definition at line 24 of file is_symmetric.hpp.

template<typename T >

bool stan::math::is_uninitialized ( T  x )

inline

Returns true if the specified variable is uninitialized.

Arithmetic types are always initialized by definition (the value is not specified).

Template Parameters

T	Type of object to test.

Parameters

x	Object to test.

Returns

true if the specified object is uninitialized.

false if input is NaN.

Definition at line 19 of file is_uninitialized.hpp.

bool stan::math::is_uninitialized ( var  x )

inline

Returns true if the specified variable is uninitialized.

This overload of the is_uninitialized() function delegates the return to the is_uninitialized() method on the specified variable.

Parameters

x	Object to test.

Returns

true if the specified object is uninitialized.

Definition at line 23 of file is_uninitialized.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

bool stan::math::is_unit_vector ( const EigVec &  theta )

inline

Return true if the vector is not a unit vector or if any element is NaN.

A valid unit vector is one where the square elements summed is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen::Matrix vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

EigMat

A type derived from EigenBase with 1 compile time row or column

Parameters

theta

Eigen vector to test

Returns

true if the vector is not a unit vector or if any element is NaN

Definition at line 28 of file is_unit_vector.hpp.

template<typename ADType , require_autodiff_t< ADType > * = nullptr>

bool stan::math::isfinite ( ADType &&  v )

inline

Return true if specified argument is finite (not infinite and not not-a-number).

Overloads std::isfinite from <cmath> for argument-dependent lookup.

Template Parameters

ADType

type of argument

Parameters

[in]

v

argument

Returns

true if argument is finite

Definition at line 21 of file isfinite.hpp.

template<typename T , typename = require_autodiff_t<T>>

bool stan::math::isinf ( const T &  v )

inline

Return true if specified argument is infinite (positive or negative).

Overloads std::isinf from <cmath> for argument-dependent lookup.

Template Parameters

ADType

type of argument

Parameters

[in]

v

argument

Returns

true if argument is infinite

Definition at line 23 of file isinf.hpp.

template<typename T , typename = require_autodiff_t<T>>

bool stan::math::isnan ( const T &  x )

inline

Return true if specified argument is not-a-number.

Overloads std::isnan from <cmath> for argument-dependent lookup.

Template Parameters

ADType

type of argument

Parameters

[in]

x

argument

Returns

true if argument is not-a-number

Definition at line 21 of file isnan.hpp.

template<typename ADType , require_autodiff_t< ADType > * = nullptr>

bool stan::math::isnormal ( ADType &&  v )

inline

Return true if specified argument is normal.

A number is normal if it is finite, non-zero and not subnormal.

Overloads std::isnormal from <cmath> for argument-dependent lookup.

Template Parameters

ADType

type of argument

Parameters

[in]

v

argument

Returns

true if argument is normal

Definition at line 21 of file isnormal.hpp.

template<typename T , typename F >

void stan::math::jacobian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 11 of file jacobian.hpp.

template<typename F >

void stan::math::jacobian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 14 of file jacobian.hpp.

template<typename F1 , typename... Args>

Eigen::VectorXd stan::math::kinsol_solve	(	const F1 &	f,
		const Eigen::VectorXd &	x,
		const double	scaling_step_tol,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		const bool	custom_jacobian,
		const int	steps_eval_jacobian,
		const int	global_line_search,
		std::ostream *const	msgs,
		const Args &...	args
	)

Return the solution to the specified algebraic system, given an initial guess.

Invokes the Kinsol solver from Sundials.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	dat	Continuous data vector for the equation system.
[in]	dat_int	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_tol	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	Maximum number of function evaluations.
[in]	custom_jacobian	If 0, use Kinsol's default to compute the jacobian for the Newton step, namely Quotient Difference (finite difference). If 1, use reverse-mode AD, unless the user specifies their own method.
[in]	J_f	A functor that computes the Jacobian for the Newton step. Defaults to reverse-mode AD.
[in]	steps_eval_jacobian	Maximum number of steps before the Jacobian gets recomputed. Note that Kinsol's default is 10. If equal to 1, the algorithm computes exact Newton steps.
[in]	global_line_search	does the solver use a global line search? If equal to KIN_NONE, no, if KIN_LINESEARCH, yes.

Returns

x_solution Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if Kinsol returns a negative flag when setting up the solver.
<code>std::domain_error</code>	if Kinsol fails to solve equation in max_num_steps iterations.
<code>std::runtime_error</code>	if Kinsol returns a negative flag that is not due to hitting max_num_steps.

Definition at line 59 of file kinsol_solve.hpp.

template<typename T >

fvar<T> stan::math::lambert_w0 ( const fvar< T > &  x )

inline

Definition at line 13 of file lambert_w.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

double stan::math::lambert_w0 ( const T &  x )

inline

Compute the Lambert W function on W0 branch for a value x.

Template Parameters

T	type of value

Parameters

x

value

Returns

value of the W0 branch of the Lambert W function for x

Exceptions

std::domain_error

if x is less than or equal to -e^(-1)

Definition at line 21 of file lambert_w.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::lambert_w0 ( const var_value< T > &  a )

inline

Return the Lambert W function on W0 branch applied to the specified variable.

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Variable argument.

Returns

the Lambert W function (W0 branch) applied to the specified argument.

Definition at line 22 of file lambert_w.hpp.

template<typename T , require_not_stan_scalar_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::lambert_w0 ( const T &  x )

inline

Vectorized version of lambert_w0().

Template Parameters

T	type of container

Parameters

x

container

Returns

value of the W0 branch of the Lambert W function for each value in x

Exceptions

std::domain_error

if x is less than or equal to -e^(-1)

Definition at line 83 of file lambert_w.hpp.

template<typename T >

fvar<T> stan::math::lambert_wm1 ( const fvar< T > &  x )

inline

Definition at line 19 of file lambert_w.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

double stan::math::lambert_wm1 ( const T &  x )

inline

Compute the Lambert W function on W-1 branch for a value x.

Template Parameters

T	type of value

Parameters

x

value

Returns

value of the W-1 branch of the Lambert W function for x

Exceptions

std::domain_error

if x is less than or equal to -e^(-1) or greater than or equal to 0

Definition at line 35 of file lambert_w.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::lambert_wm1 ( const var_value< T > &  a )

inline

Return the Lambert W function on W-1 branch applied to the specified variable.

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Variable argument.

Returns

the Lambert W function (W-1 branch) applied to the specified argument.

Definition at line 40 of file lambert_w.hpp.

template<typename T , require_not_stan_scalar_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::lambert_wm1 ( const T &  x )

inline

Vectorized version of lambert_wm1().

Template Parameters

T	type of container

Parameters

x

container

Returns

value of the W0 branch of the Lambert W function for each value in x

Exceptions

std::domain_error

if x is less than or equal to -e^(-1) or greater than or equal to 0

Definition at line 98 of file lambert_w.hpp.

template<typename T , typename L , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L > * = nullptr>

auto stan::math::lb_constrain ( T &&  x, L &&  lb  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

Specialization of lb_constrain to apply a matrix of lower bounds elementwise to each input element.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T	kernel generator expression
L	kernel generator expression

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound

Returns

constrained matrix

Template Parameters

T	A type inheriting from EigenBase.
L	A type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 32 of file lb_constrain.hpp.

template<typename T , typename L , require_all_stan_scalar_t< T, L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>

auto stan::math::lb_constrain ( const T &  x, const L &  lb  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

Specialization of lb_constrain to apply a matrix of lower bounds elementwise to each input element.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T	Scalar.
L	Scalar.

Parameters

[in]	x	Unconstrained input
[in]	lb	Lower bound

Returns

Constrained matrix

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

If the lower bound is negative infinity, this function reduces to identity_constrain(x).

Template Parameters

T	Scalar
L	Scalar

Parameters

[in]	x	Unconstrained input
[in]	lb	lower bound on constrained output

Returns

lower bound constrained value corresponding to inputs

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
L	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 37 of file lb_constrain.hpp.

template<typename T_x , typename T_lb , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb > * = nullptr, require_any_var_t< T_x, T_lb > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb > * = nullptr>

var_value<matrix_cl<double> > stan::math::lb_constrain ( T_x &&  x, T_lb &&  lb  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T_x	type of unconstrained input
T_lb	type of lower bound

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound

Returns

constrained matrix

Definition at line 39 of file lb_constrain.hpp.

template<typename T , typename L , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L > * = nullptr>

auto stan::math::lb_constrain ( const T &  x, const L &  lb, return_type_t< T, L > &  lp  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

Specialization of lb_constrain to apply a matrix of lower bounds elementwise to each input element.

Return the lower-bounded value for the specified unconstrained input and specified lower bound, incrementing the specified reference with the log absolute Jacobian determinant of the transform.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T	kernel generator expression
L	kernel generator expression

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound
[in,out]	lp	reference to log probability to increment

Returns

constrained matrix

Template Parameters

T	Scalar.
L	Scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Template Parameters

T	A type inheriting from EigenBase.
L	A type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
L	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 60 of file lb_constrain.hpp.

template<typename T_x , typename T_lb , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb > * = nullptr, require_any_var_t< T_x, T_lb > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb > * = nullptr>

var_value<matrix_cl<double> > stan::math::lb_constrain ( T_x &&  x, T_lb &&  lb, var &  lp  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T_x	type of unconstrained input
T_lb	type of lower bound

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound
[in,out]	lp	reference to log probability to increment

Returns

constrained matrix

Definition at line 75 of file lb_constrain.hpp.

template<typename T , typename L , require_eigen_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>

auto stan::math::lb_constrain ( T &&  x, L &&  lb  )

inline

Specialization of lb_constrain to apply a scalar lower bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase.
L	Scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 82 of file lb_constrain.hpp.

template<typename T , typename L , require_all_stan_scalar_t< T, L > * = nullptr, require_any_var_t< T, L > * = nullptr>

auto stan::math::lb_constrain ( const T &  x, const L &  lb, var &  lp  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

If the lower bound is negative infinity, this function reduces to identity_constrain(x).

Template Parameters

T	Scalar
L	Scalar

Parameters

[in]	x	Unconstrained input
[in]	lb	lower bound on constrained output
[in,out]	lp	reference to log probability to increment

Returns

lower bound constrained value corresponding to inputs

Definition at line 92 of file lb_constrain.hpp.

template<typename T , typename L , require_eigen_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_all_not_st_var< T, L > * = nullptr>

auto stan::math::lb_constrain ( const T &  x, const L &  lb, return_type_t< T, L > &  lp  )

inline

Specialization of lb_constrain to apply a scalar lower bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase.
L	Scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
L	Scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 100 of file lb_constrain.hpp.

template<typename T , typename L , require_matrix_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr, require_any_st_var< T, L > * = nullptr>

auto stan::math::lb_constrain ( const T &  x, const L &  lb  )

inline

Specialization of lb_constrain to apply a scalar lower bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
L	Scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 135 of file lb_constrain.hpp.

template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lb_constrain ( const std::vector< T > &  x, const L &  lb  )

inline

Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
L	A type inheriting from EigenBase or a scalar.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 153 of file lb_constrain.hpp.

template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lb_constrain ( const std::vector< T > &  x, const L &  lb, return_type_t< T, L > &  lp  )

inline

Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
L	A type inheriting from EigenBase or a standard vector.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 173 of file lb_constrain.hpp.

template<typename T , typename L >

auto stan::math::lb_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb  )

inline

Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
L	A type inheriting from EigenBase or a standard vector.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 193 of file lb_constrain.hpp.

template<typename T , typename L >

auto stan::math::lb_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb, return_type_t< T, L > &  lp  )

inline

Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
L	A type inheriting from EigenBase or a standard vector.

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 214 of file lb_constrain.hpp.

template<bool Jacobian, typename T , typename L >

auto stan::math::lb_constrain ( const T &  x, const L &  lb, return_type_t< T, L > &  lp  )

inline

Specialization of lb_constrain to apply a container of lower bounds elementwise to each input element.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
L	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound on output
[in,out]	lp	log density accumulator

Returns

lower-bound constrained value corresponding to inputs

Definition at line 243 of file lb_constrain.hpp.

template<typename T , typename L , require_not_std_vector_t< T > * = nullptr, require_stan_scalar_t< L > * = nullptr>

auto stan::math::lb_free ( T &&  y, L &&  lb  )

inline

Return the unconstrained value that produces the specified lower-bound constrained value.

Template Parameters

T	type of bounded object
L	type of lower bound

Parameters

[in]	y	input object
[in]	lb	lower bound

Returns

unconstrained value that produces the input when constrained

Exceptions

std::domain_error

if y is lower than the lower bound

Definition at line 30 of file lb_free.hpp.

template<typename T , typename L , require_all_eigen_t< T, L > * = nullptr>

auto stan::math::lb_free ( T &&  y, L &&  lb  )

inline

Return the free matrix that corresponds to the specified lower-bounded matrix with respect to the specified lower bound.

The transform is the reverse of the lb_constrain transform.

Template Parameters

T	type of matrix
L	type of lower bound

Parameters

y	constrained matrix with specified lower bound
lb	lower bound

Returns

unconstrained matrix with respect to lower bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the lower bound.

Definition at line 57 of file lb_free.hpp.

template<typename T , typename L , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lb_free ( const std::vector< T >  y, const L &  lb  )

inline

Return the free variable that corresponds to the specified lower-bounded variable with respect to the specified lower bound.

The transform is the reverse of the lb_constrain transform.

Template Parameters

T	type of constrained variable
L	type of lower bound

Parameters

y	constrained variable with specified lower bound
lb	lower bound

Returns

unconstrained variable with respect to lower bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the lower bound.

Definition at line 84 of file lb_free.hpp.

template<typename T , typename L >

auto stan::math::lb_free ( const std::vector< T >  y, const std::vector< L > &  lb  )

inline

Return the free variable that corresponds to the specified lower-bounded variable with respect to the specified lower bound.

The transform is the reverse of the lb_constrain transform.

Template Parameters

T	type of constrained variable
L	type of lower bound

Parameters

y	constrained variable with specified lower bound
lb	lower bound

Returns

unconstrained variable with respect to lower bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the lower bound.

Definition at line 107 of file lb_free.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file lbeta.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 21 of file lbeta.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 27 of file lbeta.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::lbeta ( T_a &&  a, T_b &&  b  )

inline

Return the elementwise lbeta() on two input kernel generator expression.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

elementwise lbeta()

Definition at line 32 of file lbeta.hpp.

var stan::math::lbeta ( const var &  a, const var &  b  )

inline

Returns the natural logarithm of the beta function and its gradients.

\[ \mathrm{lbeta}(a,b) = \ln\left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial a} = \psi^{\left(0\right)}\left(a\right) - \psi^{\left(0\right)}\left(a + b\right) \]

\[ \frac{\partial }{\partial b} = \psi^{\left(0\right)}\left(b\right) - \psi^{\left(0\right)}\left(a + b\right) \]

Parameters

a	var Argument
b	var Argument

Returns

Result of log beta function

Definition at line 64 of file lbeta.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::lbeta	(	const T1	a,
		const T2	b
	)

Return the log of the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

This function returns its log,

\(\log \mbox{B}(a, b) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)\).

See stan::math::lgamma() for the double-based and stan::math for the variable-based log Gamma function. This function is numerically more stable than naive evaluation via lgamma.

\[ \mbox{lbeta}(\alpha, \beta) = \begin{cases} \ln\int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \Psi(\alpha)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \Psi(\beta)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first value
T2	type of second value

Parameters

a	First value
b	Second value

Returns

Log of the beta function applied to the two values.

Definition at line 65 of file lbeta.hpp.

var stan::math::lbeta ( const var &  a, double  b  )

inline

Returns the natural logarithm of the beta function and its gradients.

\[ \mathrm{lbeta}(a,b) = \ln\left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial a} = \psi^{\left(0\right)}\left(a\right) - \psi^{\left(0\right)}\left(a + b\right) \]

Parameters

a	var Argument
b	double Argument

Returns

Result of log beta function

Definition at line 83 of file lbeta.hpp.

var stan::math::lbeta ( double  a, const var &  b  )

inline

Returns the natural logarithm of the beta function and its gradients.

\[ \mathrm{lbeta}(a,b) = \ln\left(B\left(a,b\right)\right) \]

\[ \frac{\partial }{\partial b} = \psi^{\left(0\right)}\left(b\right) - \psi^{\left(0\right)}\left(a + b\right) \]

Parameters

a	double Argument
b	var Argument

Returns

Result of log beta function

Definition at line 102 of file lbeta.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::lbeta ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the lbeta function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

lbeta function applied to the two inputs.

Definition at line 130 of file lbeta.hpp.

var stan::math::ldexp ( const var &  a, int  b  )

inline

Definition at line 19 of file ldexp.hpp.

template<typename T >

fvar<T> stan::math::ldexp ( const fvar< T > &  a, int  b  )

inline

Returns the product of a (the significand) times 2 to power b (the exponent).

Template Parameters

T	inner type of the fvar

Parameters

[in]	a	the significand
[in]	b	an integer that is the exponent

Returns

product of a times 2 to the power b

Definition at line 21 of file ldexp.hpp.

template<typename T1 , require_arithmetic_t< T1 > * = nullptr>

double stan::math::ldexp ( T1  a, int  b  )

inline

Returns the product of a (the significand) and 2 to power b (the exponent).

Template Parameters

T1	scalar type of significand

Parameters

[in]	a	the significand
[in]	b	an integer that is the exponent

Returns

product of a times 2 to the power b

Definition at line 21 of file ldexp.hpp.

template<typename T_a , typename T_b , require_all_kernel_expressions_t< T_a, T_b > * = nullptr, require_st_integral< T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::ldexp ( const var_value< T_a > &  a, T_b &&  b  )

inline

Returns the elementwise ldexp() of the input var_value<matrix_cl<double>> and kernel generator expression.

Parameters

a	input rev kernel generator expression representing significands
b	input kernel generator expression representing the integer exponents.

Returns

Elementwise ldexp() of the input argument.

Definition at line 26 of file ldexp.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

auto stan::math::ldexp ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the ldexp function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

ldexp function applied to the two inputs.

Definition at line 39 of file ldexp.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::lgamma ( const var_value< T > &  A )

inline

Returns the elementwise lgamma() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise lgamma() of the input.

Definition at line 20 of file lgamma.hpp.

template<typename T >

fvar<T> stan::math::lgamma ( const fvar< T > &  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

Argument.

Returns

natural logarithm of the gamma function of argument.

Definition at line 21 of file lgamma.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::lgamma ( const var_value< T > &  a )

inline

The log gamma function for variables (C99).

The derivative is the digamma function,

\(\frac{d}{dx} \Gamma(x) = \psi^{(0)}(x)\).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	The variable.

Returns

Log gamma of the variable.

Definition at line 24 of file lgamma.hpp.

double stan::math::lgamma ( double  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

\[ \mbox{lgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x

argument

Returns

natural logarithm of the gamma function applied to argument

Definition at line 63 of file lgamma.hpp.

double stan::math::lgamma ( int  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

Parameters

x

argument

Returns

natural logarithm of the gamma function applied to argument

Definition at line 82 of file lgamma.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::lgamma ( const T &  x )

inline

Vectorized version of lgamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of the gamma function applied to each value in x.

Exceptions

std::domain_error

if any value is a negative integer or 0.

Definition at line 119 of file lgamma.hpp.

template<typename T >

return_type_t<T> stan::math::lgamma_stirling

(

const T

x

)

Return the Stirling approximation to the lgamma function.

\[ \mbox{lgamma_stirling}(x) = \frac{1}{2} \log(2\pi) + (x-\frac{1}{2})*\log(x) - x \]

Template Parameters

T	type of value

Parameters

x

value

Returns

Stirling's approximation to lgamma(x).

Definition at line 26 of file lgamma_stirling.hpp.

template<typename T >

return_type_t<T> stan::math::lgamma_stirling_diff

(

const T

x

)

Return the difference between log of the gamma function and its Stirling approximation.

This is useful to stably compute log of ratios of gamma functions with large arguments where the Stirling approximation allows for analytic solution and the (small) differences can be added afterwards. This is for example used in the implementation of lbeta.

The function will return correct value for all arguments, but using it can lead to a loss of precision when x < lgamma_stirling_diff_useful.

\[ \mbox{lgamma_stirling_diff}(x) = \log(\Gamma(x)) - \frac{1}{2} \log(2\pi) + (x-\frac{1}{2})*\log(x) - x \]

Template Parameters

T	type of value

Parameters

x

value

Returns

Difference between lgamma(x) and its Stirling approximation.

Definition at line 43 of file lgamma_stirling_diff.hpp.

std::vector<int> stan::math::linspaced_array ( int  K, double  low, double  high  )

inline

Return an array of linearly spaced elements.

This produces an array from low to high (inclusive) with elements spaced as (high - low) / (K - 1). For K=1, the array will contain the high value; for K=0 it returns an empty array.

Parameters

K	size of the array
low	smallest value
high	largest value

Returns

An array of size K with elements linearly spaced between low and high.

Exceptions

std::domain_error

if K is negative, if low is nan or infinite, if high is nan or infinite, or if high is less than low.

Definition at line 26 of file linspaced_array.hpp.

std::vector<int> stan::math::linspaced_int_array ( int  K, int  low, int  high  )

inline

Return an array of linearly spaced integers.

This produces an array from low to high (inclusive). If high - low is greater or equal to K - 1, then the integers are evenly spaced. If it is not possible to get from low to high with a multiple of an integer, high is lowered until this is possible.

If K - 1 is greater than high - low, then integers are repeated. For instance, low, low, low + 1, low + 1, .... high is lowered until K - 1 is a multiple of high - low

For K = 1, the array will contain the high value. For K = 0 it returns an empty array.

Parameters

K	size of the array
low	smallest value
high	largest value

Returns

An array of size K with elements linearly spaced between low and high.

Exceptions

std::domain_error

if K is negative or high is less than low.

Definition at line 33 of file linspaced_int_array.hpp.

auto stan::math::linspaced_row_vector ( int  K, double  low, double  high  )

inline

Return a row vector of linearly spaced elements.

This produces a row vector from low to high (inclusive) with elements spaced as (high - low) / (K - 1). For K=1, the vector will contain the high value; for K=0 it returns an empty vector.

Parameters

K	size of the row vector
low	smallest value
high	largest value

Returns

A row vector of size K with elements linearly spaced between low and high.

Exceptions

std::domain_error

if K is negative, if low is nan or infinite, if high is nan or infinite, or if high is less than low.

Definition at line 25 of file linspaced_row_vector.hpp.

auto stan::math::linspaced_vector ( int  K, double  low, double  high  )

inline

Return a vector of linearly spaced elements.

This produces a vector from low to high (inclusive) with elements spaced as (high - low) / (K - 1). For K=1, the vector will contain the high value; for K=0 it returns an empty vector.

Parameters

K	size of the vector
low	smallest value
high	largest value

Returns

A vector of size K with elements linearly spaced between low and high.

Exceptions

std::domain_error

if K is negative, if low is nan or infinite, if high is nan or infinite, or if high is less than low.

Definition at line 25 of file linspaced_vector.hpp.

template<bool propto, typename T_covar , typename T_shape >

return_type_t<T_covar, T_shape> stan::math::lkj_corr_cholesky_lpdf	(	const T_covar &	L,
		const T_shape &	eta
	)

Definition at line 18 of file lkj_corr_cholesky_lpdf.hpp.

template<typename T_covar , typename T_shape >

return_type_t<T_covar, T_shape> stan::math::lkj_corr_cholesky_lpdf ( const T_covar &  L, const T_shape &  eta  )

inline

Definition at line 56 of file lkj_corr_cholesky_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::lkj_corr_cholesky_rng ( size_t  K, double  eta, RNG &  rng  )

inline

Definition at line 13 of file lkj_corr_cholesky_rng.hpp.

template<bool propto, typename T_y , typename T_shape >

return_type_t<T_y, T_shape> stan::math::lkj_corr_lpdf	(	const T_y &	y,
		const T_shape &	eta
	)

Definition at line 49 of file lkj_corr_lpdf.hpp.

template<typename T_y , typename T_shape >

return_type_t<T_y, T_shape> stan::math::lkj_corr_lpdf ( const T_y &  y, const T_shape &  eta  )

inline

Definition at line 80 of file lkj_corr_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_eigen_matrix_dynamic_t< T_y > * = nullptr, require_all_eigen_col_vector_t< T_loc, T_scale > * = nullptr>

return_type_t< T_y, T_loc, T_scale, T_shape > stan::math::lkj_cov_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	eta
	)

Deprecated:

Deprecated:

Definition at line 21 of file lkj_cov_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::lkj_cov_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma, const T_shape &  eta  )

inline

Deprecated:

Definition at line 100 of file lkj_cov_lpdf.hpp.

template<typename T >

fvar<return_type_t<T, int> > stan::math::lmgamma ( int  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file lmgamma.hpp.

var stan::math::lmgamma ( int  a, const var &  b  )

inline

Definition at line 27 of file lmgamma.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

return_type_t<T> stan::math::lmgamma ( int  k, T  x  )

inline

Return the natural logarithm of the multivariate gamma function with the specified dimensions and argument.

The multivariate gamma function \(\Gamma_k(x)\) for dimensionality \(k\) and argument \(x\) is defined by

\(\Gamma_k(x) = \pi^{k(k-1)/4} \, \prod_{j=1}^k \Gamma(x + (1 - j)/2)\),

where \(\Gamma()\) is the gamma function.

\[ \mbox{lmgamma}(n, x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma_n(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lmgamma}(n, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \ln\Gamma_n(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \ln\Gamma_n(x) = \pi^{n(n-1)/4} \, \prod_{j=1}^n \Gamma(x + (1 - j)/2) \]

\[ \frac{\partial \, \ln\Gamma_n(x)}{\partial x} = \sum_{j=1}^n \Psi(x + (1 - j) / 2) \]

Template Parameters

T	type of scalar

Parameters

k	Number of dimensions.
x	Function argument.

Returns

Natural log of the multivariate gamma function.

Definition at line 55 of file lmgamma.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::lmgamma ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the natural log of the multivariate gamma function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

Natural log of the multivariate gamma function applied to the two inputs.

Definition at line 73 of file lmgamma.hpp.

template<typename T >

fvar<T> stan::math::lmultiply ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 13 of file lmultiply.hpp.

template<typename T >

fvar<T> stan::math::lmultiply ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 20 of file lmultiply.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::lmultiply ( const T1  a, const T2  b  )

inline

Return the first argument times the log of the second argument.

The result is 0 if both arguments are 0. The funcgtion is defined by lmultiply(x, y) = x * log(y) if x or y is non-zero and lmultiply(0, 0) = 0 otherwise.

Template Parameters

T1	type of the first argument
T2	type of the second argument

Parameters

a	first argument
b	second argument

Returns

the first argument times the log of the second argument

Definition at line 25 of file lmultiply.hpp.

template<typename T >

fvar<T> stan::math::lmultiply ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 26 of file lmultiply.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::lmultiply ( T_a &&  a, T_b &&  b  )

inline

Returns the elementwise lmultiply() of the input.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Elementwise lmultiply() of the input.

Definition at line 28 of file lmultiply.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::lmultiply ( const T1 &  a, const T2 &  b  )

inline

Return the result of applying lmultiply to the arguments elementwise, with broadcasting if one of the arguments is a scalar.

Return the product a * log(b).

Return the elementwise product a * log(b).

At least one of the arguments must be a container.

Template Parameters

T1	type of the first argument
T2	type of the second argument

Parameters

a	first argument
b	second argument

Returns

result of applying lmultiply to the arguments

Both T1 and T2 are matrices, and one of T1 or T2 must be a var_value

Template Parameters

T1	Type of first argument
T2	Type of second argument

Parameters

a	First argument
b	Second argument

Returns

elementwise product of a and log(b)

Template Parameters

T1	Type of matrix argument
T2	Type of scalar argument

Parameters

a	Matrix argument
b	Scalar argument

Returns

Product of a and log(b)

Template Parameters

T1	Type of scalar argument
T2	Type of matrix argument

Parameters

a	Scalar argument
b	Matrix argument

Returns

Product of a and log(b)

Definition at line 46 of file lmultiply.hpp.

var stan::math::lmultiply ( const var &  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to a is log(b). The partial derivative with respect to b is a/b.

Parameters

a	First variable.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 57 of file lmultiply.hpp.

var stan::math::lmultiply ( const var &  a, double  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to a is log(b).

Parameters

a	First variable.
b	Second scalar.

Returns

Value of a*log(b)

Definition at line 70 of file lmultiply.hpp.

var stan::math::lmultiply ( double  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to b is a/b.

Parameters

a	First scalar.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 83 of file lmultiply.hpp.

template<typename T >

fvar<T> stan::math::log ( const fvar< T > &  x )

inline

Definition at line 15 of file log.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log ( const var_value< T > &  A )

inline

Returns the elementwise log() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log() of the input.

Definition at line 20 of file log.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::log ( const std::complex< fvar< T >> &  z )

inline

Return the natural logarithm (base e) of the specified complex argument.

Template Parameters

T	autodiff value type

Parameters

z	complex argument

Returns

natural logarithm of argument

Definition at line 32 of file log.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::log ( const Container &  x )

inline

Return the elementwise natural log of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

Container

type of container

Parameters

[in]

x

container

Returns

Elementwise application of natural log to the argument.

Definition at line 50 of file log.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log ( const var_value< T > &  a )

inline

Return the natural log of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \log x = \frac{1}{x}\).

\[ \mbox{log}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \ln(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Variable whose log is taken.

Returns

Natural log of variable.

Definition at line 50 of file log.hpp.

std::complex<var> stan::math::log ( const std::complex< var > &  z )

inline

Return the natural logarithm (base e) of the specified complex argument.

Parameters

z	complex argument

Returns

natural logarithm of argument

Definition at line 63 of file log.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::log ( const Container &  x )

inline

Version of log() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Natural log of each variable in the container.

Definition at line 64 of file log.hpp.

template<typename T >

fvar<T> stan::math::log10 ( const fvar< T > &  x )

inline

Definition at line 15 of file log10.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log10 ( const var_value< T > &  A )

inline

Returns the elementwise log10() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log10() of the input.

Definition at line 20 of file log10.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::log10 ( const std::complex< fvar< T >> &  z )

inline

Return the base 10 logarithm of the specified complex number.

Template Parameters

T	autodiff value type

Parameters

z	complex argument

Returns

base 10 log of argument

Definition at line 33 of file log10.hpp.

template<typename Container , require_not_var_matrix_t< Container > * = nullptr, require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::log10 ( const Container &  x )

inline

Vectorized version of log10().

Template Parameters

Container

type of container

Parameters

x

container

Returns

Log base-10 applied to each value in x.

Definition at line 41 of file log10.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log10 ( const var_value< T > &  a )

inline

Return the base 10 log of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \log_{10} x = \frac{1}{x \log 10}\).

\[ \mbox{log10}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \log_{10}(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log10}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x \ln10} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	Variable whose log is taken.

Returns

Base 10 log of variable.

Definition at line 49 of file log10.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::log10 ( const Container &  x )

inline

Version of log10() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Log base-10 of each variable in the container.

Definition at line 55 of file log10.hpp.

std::complex<var> stan::math::log10 ( const std::complex< var > &  z )

inline

Return the base 10 logarithm of the specified complex number.

Parameters

z	complex argument

Returns

base 10 log of argument

Definition at line 62 of file log10.hpp.

static constexpr double stan::math::log10 ( )

inlinestatic

Returns the natural logarithm of ten.

Returns

Natural logarithm of ten.

Definition at line 228 of file constants.hpp.

template<typename T >

fvar<T> stan::math::log1m ( const fvar< T > &  x )

inline

Definition at line 12 of file log1m.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log1m ( const var_value< T > &  A )

inline

Returns the elementwise log1m() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log1m() of the input.

Definition at line 20 of file log1m.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log1m ( const var_value< T > &  a )

inline

The log (1 - x) function for variables.

The derivative is given by

\(\frac{d}{dx} \log (1 - x) = -\frac{1}{1 - x}\).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	The variable.

Returns

The variable representing log of 1 minus the variable.

Definition at line 23 of file log1m.hpp.

double stan::math::log1m ( double  x )

inline

Return the natural logarithm of one minus the specified value.

The main use of this function is to cut down on intermediate values during algorithmic differentiation.

\[ \mbox{log1m}(x) = \begin{cases} \ln(1-x) & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m}(x)}{\partial x} = \begin{cases} -\frac{1}{1-x} & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

[in]

x

Argument.

Returns

Natural log of one minus the argument.

Exceptions

<code>std::domain_error</code>	If the argument is greater than 1.
<code>std::overflow_error</code>	If the computation overflows.

Definition at line 42 of file log1m.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::log1m ( const T &  x )

inline

Vectorized version of log1m().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of 1 minus each value in x.

Definition at line 73 of file log1m.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log1m_exp ( const var_value< T > &  A )

inline

Returns the elementwise log1m_exp() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log1m_exp() of the input.

Definition at line 20 of file log1m_exp.hpp.

template<typename T >

fvar<T> stan::math::log1m_exp ( const fvar< T > &  x )

inline

Return the natural logarithm of one minus the exponentiation of the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

Argument.

Returns

log of one minus the exponentiation of the argument.

Definition at line 23 of file log1m_exp.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log1m_exp ( const var_value< T > &  x )

inline

Return the log of 1 minus the exponential of the specified variable.

The derivative of log(1 - exp(x)) with respect to x is -1 / expm1(-x).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

[in]

x

Argument.

Returns

Natural logarithm of one minus the exponential of the argument.

Definition at line 25 of file log1m_exp.hpp.

double stan::math::log1m_exp ( double  a )

inline

Calculates the natural logarithm of one minus the exponential of the specified value without overflow,.

log1m_exp(x) = log(1-exp(x))

This function is only defined for x < 0

\[ \mbox{log1m\_exp}(x) = \begin{cases} \ln(1-\exp(x)) & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{1-\exp(x)} & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

[in]

a

Argument.

Returns

natural logarithm of one minus the exponential of the argument.

Definition at line 47 of file log1m_exp.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::log1m_exp ( const T &  x )

inline

Vectorized version of log1m_exp().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of (1 - exp()) applied to each value in x.

Definition at line 83 of file log1m_exp.hpp.

template<typename T >

fvar<T> stan::math::log1m_inv_logit ( const fvar< T > &  x )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 21 of file log1m_inv_logit.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log1m_inv_logit ( const var_value< T > &  A )

inline

Returns the elementwise log1m_inv_logit() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log1m_inv_logit() of the input.

Definition at line 21 of file log1m_inv_logit.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log1m_inv_logit ( const var_value< T > &  u )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 21 of file log1m_inv_logit.hpp.

double stan::math::log1m_inv_logit ( double  u )

inline

Returns the natural logarithm of 1 minus the inverse logit of the specified argument.

\[ \mbox{log1m\_inv\_logit}(x) = \begin{cases} -\ln(\exp(x)+1) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m\_inv\_logit}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{\exp(x)+1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 36 of file log1m_inv_logit.hpp.

double stan::math::log1m_inv_logit ( int  u )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 51 of file log1m_inv_logit.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

apply_scalar_unary<log1m_inv_logit_fun, T>::return_t stan::math::log1m_inv_logit ( const T &  x )

inline

Return the elementwise application of log1m_inv_logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise log1m_inv_logit of members of container.

Definition at line 86 of file log1m_inv_logit.hpp.

template<typename T >

fvar<T> stan::math::log1p ( const fvar< T > &  x )

inline

Definition at line 12 of file log1p.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log1p ( const var_value< T > &  A )

inline

Returns the elementwise log1p() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log1p() of the input.

Definition at line 20 of file log1p.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log1p ( const var_value< T > &  a )

inline

The log (1 + x) function for variables (C99).

The derivative is given by

\(\frac{d}{dx} \log (1 + x) = \frac{1}{1 + x}\).

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	The variable.

Returns

The log of 1 plus the variable.

Definition at line 23 of file log1p.hpp.

double stan::math::log1p ( double  x )

inline

Return the natural logarithm of one plus the specified value.

\[ \mbox{log1p}(x) = \log(1 + x) \]

This version is more stable for arguments near zero than the direct definition. If x == -1, log1p(x) is defined to be negative infinity.

Parameters

[in]

x

Argument.

Returns

Natural log of one plus the argument.

Exceptions

std::domain_error

If argument is less than -1.

Definition at line 29 of file log1p.hpp.

double stan::math::log1p ( int  x )

inline

Return the natural logarithm of one plus the specified argument.

This version is required to disambiguate log1p(int).

Parameters

[in]

x

Argument.

Returns

Natural logarithm of one plus the argument.

Exceptions

std::domain_error

If argument is less than -1.

Definition at line 47 of file log1p.hpp.

template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::log1p ( const T &  x )

inline

Return the elementwise application of log1p() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise log1p of members of container.

Definition at line 82 of file log1p.hpp.

template<typename T >

fvar<T> stan::math::log1p_exp ( const fvar< T > &  x )

inline

Definition at line 13 of file log1p_exp.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log1p_exp ( const var_value< T > &  a )

inline

Return the log of 1 plus the exponential of the specified variable.

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	The variable.

Definition at line 19 of file log1p_exp.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log1p_exp ( const var_value< T > &  A )

inline

Returns the elementwise log1p_exp() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log1p_exp() of the input.

Definition at line 20 of file log1p_exp.hpp.

double stan::math::log1p_exp ( double  a )

inline

Calculates the log of 1 plus the exponential of the specified value without overflow.

This function is related to other special functions by:

log1p_exp(x)

= log1p(exp(a))

= log(1 + exp(x))

= log_sum_exp(0, x).

\[ \mbox{log1p\_exp}(x) = \begin{cases} \ln(1+\exp(x)) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1p\_exp}(x)}{\partial x} = \begin{cases} \frac{\exp(x)}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 45 of file log1p_exp.hpp.

template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::log1p_exp ( const T &  x )

inline

Vectorized version of log1p_exp().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of (1 + exp()) applied to each value in x.

Definition at line 78 of file log1p_exp.hpp.

double stan::math::log2 ( )

inline

Return natural logarithm of two.

Returns

Natural logarithm of two.

Definition at line 17 of file log2.hpp.

template<typename T >

fvar<T> stan::math::log2 ( const fvar< T > &  x )

inline

Return the base two logarithm of the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

argument

Returns

base two logarithm of argument

Definition at line 20 of file log2.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log2 ( const var_value< T > &  A )

inline

Returns the elementwise log2() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log2() of the input.

Definition at line 20 of file log2.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::log2 ( const var_value< T > &  a )

inline

Returns the base 2 logarithm of the specified variable (C99).

See log2() for the double-based version.

The derivative is

\(\frac{d}{dx} \log_2 x = \frac{1}{x \log 2}\).

\[ \mbox{log2}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \log_2(x) & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log2}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \frac{1}{x\ln2} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

a	The variable.

Returns

Base 2 logarithm of the variable.

Definition at line 44 of file log2.hpp.

template<typename T , require_not_var_matrix_t< T > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::log2 ( const T &  x )

inline

Return the elementwise application of log2() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise log2 of container elements

Definition at line 49 of file log2.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

var stan::math::log_determinant ( const T &  m )

inline

Definition at line 14 of file log_determinant.hpp.

template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>

value_type_t< EigMat > stan::math::log_determinant ( const EigMat &  m )

inline

Returns the log absolute determinant of the specified square matrix.

Template Parameters

EigMat

type of the matrix

Parameters

m	Specified matrix.

Returns

log absolute determinant of the matrix.

Exceptions

std::domain_error

if matrix is not square.

Definition at line 16 of file log_determinant.hpp.

template<typename T , require_not_rev_matrix_t< T > * = nullptr>

value_type_t<T> stan::math::log_determinant_ldlt ( LDLT_factor< T > &  A )

inline

Returns log(abs(det(A))) given a LDLT_factor of A.

Template Parameters

T	type of elements in the LDLT_factor

Parameters

A	LDLT_factor

Returns

the log(abs(det(A))

Definition at line 17 of file log_determinant_ldlt.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

var stan::math::log_determinant_ldlt

(

LDLT_factor< T > &

A

)

Returns the log det of the matrix whose LDLT factorization is given.

Template Parameters

T	Type of matrix for LDLT factor

Parameters

A	an LDLT_factor

Returns

ln(det(A))

Definition at line 21 of file log_determinant_ldlt.hpp.

template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>

var stan::math::log_determinant_spd ( const EigMat &  m )

inline

Returns the log det of a symmetric, positive-definite matrix.

Template Parameters

EigMat

Type of the matrix

Parameters

m	a symmetric, positive-definite matrix

Returns

The log determinant of the specified matrix

Definition at line 24 of file log_determinant_spd.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>

value_type_t<EigMat> stan::math::log_determinant_spd ( const EigMat &  m )

inline

Returns the log absolute determinant of the specified square matrix.

Template Parameters

T	type of the matrix

Parameters

m	specified matrix

Returns

log absolute determinant of the matrix

Exceptions

std::domain_error

if matrix is not square and symmetric

Definition at line 25 of file log_determinant_spd.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

var stan::math::log_determinant_spd ( const T &  m )

inline

Returns the log det of a symmetric, positive-definite matrix.

Template Parameters

EigMat

Type of the matrix

Parameters

m	a symmetric, positive-definite matrix

Returns

The log determinant of the specified matrix

Definition at line 74 of file log_determinant_spd.hpp.

template<typename T >

fvar<T> stan::math::log_diff_exp ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file log_diff_exp.hpp.

template<typename T1 , typename T2 , require_arithmetic_t< T1 > * = nullptr>

fvar<T2> stan::math::log_diff_exp ( const T1 &  x1, const fvar< T2 > &  x2  )

inline

Definition at line 27 of file log_diff_exp.hpp.

template<typename T_x , typename T_y , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_any_var_t< T_x, T_y > * = nullptr, require_any_not_stan_scalar_t< T_x, T_y > * = nullptr>

var_value<matrix_cl<double> > stan::math::log_diff_exp ( T_x &&  x, T_y &&  y  )

inline

Returns the natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y.

Template Parameters

T_x	type of x argument
T_y	type of y argument

Parameters

x	first argument
y	second argument

Returns

Result the natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y.

Definition at line 36 of file log_diff_exp.hpp.

template<typename T1 , typename T2 , require_arithmetic_t< T2 > * = nullptr>

fvar<T1> stan::math::log_diff_exp ( const fvar< T1 > &  x1, const T2 &  x2  )

inline

Definition at line 38 of file log_diff_exp.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::log_diff_exp ( const T1  x, const T2  y  )

inline

The natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y.

This function is only defined for x >= y

\[ \mbox{log\_diff\_exp}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x < y\\ \ln(\exp(x)-\exp(y)) & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ \frac{\exp(x)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ -\frac{\exp(y)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of the first argument
T2	type of the second argument

Parameters

x	first argument
y	second argument

Definition at line 51 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const var &  a, const var &  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 56 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const var &  a, double  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 67 of file log_diff_exp.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::log_diff_exp ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the log_diff_exp function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

log_diff_exp function applied to the two inputs.

Definition at line 69 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( double  a, const var &  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 78 of file log_diff_exp.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x, const fvar< T > &  n  )

inline

Definition at line 14 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( double  x, const fvar< T > &  n  )

inline

Definition at line 21 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x, double  n  )

inline

Definition at line 27 of file log_falling_factorial.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::log_falling_factorial ( const T1  x, const T2  n  )

inline

Return the natural log of the falling factorial of the specified arguments.

\[ \mbox{log\_falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ -\Psi(n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

[in]	x	First argument
[in]	n	Second argument

Returns

log of falling factorial of arguments

Exceptions

std::domain_error

if the first argument is not positive

Definition at line 55 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var &  a, double  b  )

inline

Definition at line 61 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var &  a, const var &  b  )

inline

Definition at line 65 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( double  a, const var &  b  )

inline

Definition at line 69 of file log_falling_factorial.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::log_falling_factorial ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the log_falling_factorial function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

log_falling_factorial function applied to the two inputs.

Definition at line 75 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit ( const fvar< T > &  x )

inline

Definition at line 13 of file log_inv_logit.hpp.

var stan::math::log_inv_logit ( const var &  u )

inline

Return the natural logarithm of the inverse logit of the specified argument.

Parameters

u

argument

Returns

log inverse logit of the argument

Definition at line 19 of file log_inv_logit.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log_inv_logit ( const var_value< T > &  A )

inline

Returns the elementwise log_inv_logit() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise log_inv_logit() of the input.

Definition at line 20 of file log_inv_logit.hpp.

double stan::math::log_inv_logit ( double  u )

inline

Returns the natural logarithm of the inverse logit of the specified argument.

\[ \mbox{log\_inv\_logit}(x) = \begin{cases} \ln\left(\frac{1}{1+\exp(-x)}\right)& \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_inv\_logit}(x)}{\partial x} = \begin{cases} \frac{1}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log of the inverse logit of argument

Definition at line 34 of file log_inv_logit.hpp.

double stan::math::log_inv_logit ( int  u )

inline

Returns the natural logarithm of the inverse logit of the specified argument.

Parameters

u

argument

Returns

log of the inverse logit of argument

Definition at line 49 of file log_inv_logit.hpp.

template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::log_inv_logit ( const T &  x )

inline

Return the elementwise application of log_inv_logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise log_inv_logit of members of container

Definition at line 83 of file log_inv_logit.hpp.

template<typename T_x , typename T_y , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_any_var_t< T_x, T_y > * = nullptr, require_any_not_stan_scalar_t< T_x, T_y > * = nullptr>

var_value<matrix_cl<double> > stan::math::log_inv_logit_diff ( T_x &&  x, T_y &&  y  )

inline

Returns the natural logarithm of the difference of the inverse logits of the specified arguments.

Template Parameters

T_x	type of x argument
T_y	type of y argument

Parameters

x	first argument
y	second argument

Returns

Result of log difference of inverse logits of arguments.

Definition at line 34 of file log_inv_logit_diff.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::log_inv_logit_diff ( const T1 &  x, const T2 &  y  )

inline

Returns the natural logarithm of the difference of the inverse logits of the specified arguments.

\[ \mathrm{log\_inv\_logit\_diff}(x,y) = \ln\left(\frac{1}{1+\exp(-x)}-\frac{1}{1+\exp(-y)}\right) \]

\[ \frac{\partial }{\partial x} = -\frac{e^x}{e^y-e^x}-\frac{e^x}{e^x+1} \]

\[ \frac{\partial }{\partial x} = -\frac{e^y}{e^x-e^y}-\frac{e^y}{e^y+1} \]

Template Parameters

T1	type of x argument
T2	type of y argument

Parameters

x	first argument
y	second argument

Returns

Result of log difference of inverse logits of arguments.

Definition at line 36 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Returns fvar with the natural logarithm of the difference of the inverse logits of the specified arguments and its gradients.

\[ \mathrm{log\_inv\_logit\_diff}(x,y) = \ln\left(\frac{1}{1+\exp(-x)}-\frac{1}{1+\exp(-y)}\right) \]

\[ \frac{\partial }{\partial x} = -\frac{e^x}{e^y-e^x}-\frac{e^x}{e^x+1} \]

\[ \frac{\partial }{\partial x} = -\frac{e^y}{e^x-e^y}-\frac{e^y}{e^y+1} \]

Template Parameters

T	inner type of the fvar

Parameters

x	Argument.
y	Argument.

Returns

Fvar with result of log difference of inverse logits of arguments and gradients.

Definition at line 38 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( const fvar< T > &  x, double  y  )

inline

Definition at line 46 of file log_inv_logit_diff.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::log_inv_logit_diff ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the log_inv_logit_diff function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

log_inv_logit_diff function applied to the two inputs.

Definition at line 51 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( double  x, const fvar< T > &  y  )

inline

Definition at line 52 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( const var &  a, double  b  )

inline

Definition at line 71 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( const var &  a, const var &  b  )

inline

Definition at line 75 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( double  a, const var &  b  )

inline

Definition at line 79 of file log_inv_logit_diff.hpp.

template<typename T_theta_cl , typename T_lambda_cl , require_all_prim_or_rev_kernel_expression_t< T_theta_cl, T_lambda_cl > * = nullptr>

auto stan::math::log_mix ( const T_theta_cl &  theta, const T_lambda_cl &  lambda  )

inline

Return the log mixture density with specified mixing proportions and log densities.

\[ \frac{\partial }{\partial p_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

\[ \frac{\partial }{\partial d_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Template Parameters

T_theta	Type of theta.
T_lam	Type of lambda.

Parameters

theta	std/row/col vector of mixing proportions in [0, 1].
lambda	std/row/col vector of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 39 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 , require_all_arithmetic_t< T_theta, T_lambda1, T_lambda2 > * = nullptr>

double stan::math::log_mix ( T_theta  theta, T_lambda1  lambda1, T_lambda2  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \lambda_1 + (1 - \theta) \lambda_2 \right). \]

Template Parameters

T_theta	type of mixing proportion - must be an arithmetic type
T_lambda1	type of first log density - must be an arithmetic type
T_lambda2	type of second log density - must be an arithmetic type

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 41 of file log_mix.hpp.

template<typename T_theta , typename T_lam , require_any_vector_t< T_theta, T_lam > * = nullptr>

return_type_t<T_theta, T_lam> stan::math::log_mix	(	const T_theta &	theta,
		const T_lam &	lambda
	)

Return the log mixture density with specified mixing proportions and log densities.

\[ \frac{\partial }{\partial p_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

\[ \frac{\partial }{\partial d_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Template Parameters

T_theta	Type of theta. This can be a scalar, std vector or row/column vector.
T_lam	Type of lambda. This can be a scalar, std vector or row/column vector.

Parameters

theta	std/row/col vector of mixing proportions in [0, 1].
lambda	std/row/col vector of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 77 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 , require_any_var_t< T_theta, T_lambda1, T_lambda2 > * = nullptr>

return_type_t<T_theta, T_lambda1, T_lambda2> stan::math::log_mix ( const T_theta &  theta, const T_lambda1 &  lambda1, const T_lambda2 &  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters

T_theta	theta scalar type.
T_lambda1	lambda1 scalar type.
T_lambda2	lambda2 scalar type.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 79 of file log_mix.hpp.

template<typename T >

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, const fvar< T > &  lambda1, const fvar< T > &  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters

T	inner type of the fvar

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 98 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, const fvar< T > &  lambda1, P  lambda2  )

inline

Definition at line 118 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, P  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 136 of file log_mix.hpp.

template<typename T_theta , typename T_lam , require_vector_t< T_lam > * = nullptr>

return_type_t<T_theta, std::vector<T_lam> > stan::math::log_mix	(	const T_theta &	theta,
		const std::vector< T_lam > &	lambda
	)

Return the log mixture density given specified mixing proportions and array of log density vectors.

\[ \frac{\partial }{\partial p_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right)+ \log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m}+ \frac{e^{f_x}}{e^{f_1}p_1+\cdot\cdot\cdot+e^{f_m}p_m} \]

\[ \frac{\partial }{\partial d_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) +\log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Template Parameters

T_theta	Type of theta. This can be a scalar, std vector or row/column vector
T_lam	Type of vector in std vector lambda. This can be std vector or row/column vector.

Parameters

theta	std/row/col vector of mixing proportions in [0, 1].
lambda	std vector containing std/row/col vectors of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 147 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > * = nullptr>

fvar<T> stan::math::log_mix ( P  theta, const fvar< T > &  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 154 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, P1  lambda1, P2  lambda2  )

inline

Definition at line 173 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>

fvar<T> stan::math::log_mix ( P1  theta, const fvar< T > &  lambda1, P2  lambda2  )

inline

Definition at line 189 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > * = nullptr>

fvar<T> stan::math::log_mix ( P1  theta, P2  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 205 of file log_mix.hpp.

void stan::math::log_mix_partial_helper ( double  theta_val, double  lambda1_val, double  lambda2_val, double &  one_m_exp_lam2_m_lam1, double &  one_m_t_prod_exp_lam2_m_lam1, double &  one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1  )

inline

Definition at line 24 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>

void stan::math::log_mix_partial_helper ( const T_theta &  theta, const T_lambda1 &  lambda1, const T_lambda2 &  lambda2, promote_args_t< T_theta, T_lambda1, T_lambda2 >(&)  partials_array[N]  )

inline

Definition at line 28 of file log_mix.hpp.

template<typename T1 , typename T2 , require_all_stan_scalar_t< T1, T2 > * = nullptr>

return_type_t<T1, T2, double> stan::math::log_modified_bessel_first_kind ( const T1  v, const T2  z  )

inline

Definition at line 47 of file log_modified_bessel_first_kind.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::log_modified_bessel_first_kind ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the log_modified_bessel_first_kind function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

log_modified_bessel_first_kind function applied to the two inputs.

Definition at line 234 of file log_modified_bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x, const fvar< T > &  n  )

inline

Definition at line 13 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x, double  n  )

inline

Definition at line 20 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( double  x, const fvar< T > &  n  )

inline

Definition at line 26 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var &  a, double  b  )

inline

Definition at line 43 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var &  a, const var &  b  )

inline

Definition at line 47 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( double  a, const var &  b  )

inline

Definition at line 51 of file log_rising_factorial.hpp.

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::log_rising_factorial ( const T1 &  x, const T2 &  n  )

inline

Return the natural logarithm of the rising factorial from the first specified argument to the second.

\[ \mbox{log\_rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) - \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument x
T2	type of second argument n

Parameters

[in]	x	first argument
[in]	n	second argument

Returns

natural logarithm of the rising factorial from x to n

Exceptions

std::domain_error

if the first argument is not positive

Definition at line 53 of file log_rising_factorial.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::log_rising_factorial ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the log_rising_factorial function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

log_rising_factorial function applied to the two inputs.

Definition at line 73 of file log_rising_factorial.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

matrix_cl<double> stan::math::log_softmax ( const T &  a )

inline

Return log of the softmax of the specified vector.

Template Parameters

T	type of the vector

Parameters

[in]

a

vector to transform

Returns

log of the softmax

Definition at line 23 of file log_softmax.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::log_softmax ( const var_value< T > &  A )

inline

Return log of the softmax of the specified vector.

Template Parameters

T	type of the vector

Parameters

[in]

A

vector

Returns

log of the softmax

Definition at line 23 of file log_softmax.hpp.

template<typename T , require_vector_st< is_fvar, T > * = nullptr>

auto stan::math::log_softmax ( const T &  x )

inline

Return the log softmax of the specified vector or container of vectors.

Return the log softmax of the specified std::vector or std::vector of containers.

Return the log softmax of the specified vector.

Template Parameters

T	Type of input vector or matrix.

Parameters

[in]

x

Unconstrained input vector.

Returns

Softmax of the input.

Exceptions

std::domain_error

If the input vector is size 0.

Template Parameters

T	type of input

Parameters

x

input

Returns

softmax of the input

Exceptions

std::domain_error

if the input size is 0

Definition at line 23 of file log_softmax.hpp.

template<typename Container , require_st_arithmetic< Container > * = nullptr, require_container_t< Container > * = nullptr>

auto stan::math::log_softmax ( const Container &  x )

inline

Return the natural logarithm of the softmax of the specified vector.

\( \log \mbox{softmax}(y) \ = \ y - \log \sum_{k=1}^K \exp(y_k) \ = \ y - \mbox{log\_sum\_exp}(y). \)

For the log softmax function, the entries in the Jacobian are \( \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] = \left\{ \begin{array}{ll} 1 - \mbox{softmax}(y)[m] & \mbox{ if } m = k, \mbox{ and} \\[6pt] \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. \)

Template Parameters

Container

type of input vector to transform

Parameters

[in]

x

vector to transform

Returns

log unit simplex result of the softmax transform of the vector.

Definition at line 44 of file log_softmax.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 18 of file log_sum_exp.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::log_sum_exp ( const var_value< T > &  A )

inline

Return the log of the sum of the exponentiated values of the specified matrix of values.

Template Parameters

T	type of the vector

Parameters

[in]

A

vector

Returns

the log of the sum of the exponentiated vector values

Definition at line 24 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 26 of file log_sum_exp.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

double stan::math::log_sum_exp ( const T &  a )

inline

Return the log of the sum of the exponentiated values of the specified matrix of values.

The matrix may be a full matrix, a vector, a row vector.

The function is defined as follows to prevent overflow in exponential calculations.

\(\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))\).

Template Parameters

T	type of input vector or matrix

Parameters

[in]

a

matrix of specified values

Returns

the log of the sum of the exponentiated vector values

Definition at line 31 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 35 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const var &  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 45 of file log_sum_exp.hpp.

template<typename T1 , typename T2 , require_all_not_st_var< T1, T2 > * = nullptr>

return_type_t<T1, T2> stan::math::log_sum_exp ( const T2 &  a, const T1 &  b  )

inline

Calculates the log sum of exponentials without overflow.

\(\log (\exp(a) + \exp(b)) = m + \log(\exp(a-m) + \exp(b-m))\),

where \(m = max(a, b)\).

\[ \mbox{log\_sum\_exp}(x, y) = \begin{cases} \ln(\exp(x)+\exp(y)) & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial x} = \begin{cases} \frac{\exp(x)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial y} = \begin{cases} \frac{\exp(y)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of the first variable
T2	type of the second variable

Parameters

a	the first variable
b	the second variable

Definition at line 51 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const var &  a, double  b  )

inline

Returns the log sum of exponentials.

Definition at line 51 of file log_sum_exp.hpp.

template<typename T , require_container_st< is_fvar, T > * = nullptr>

auto stan::math::log_sum_exp ( const T &  x )

inline

Return the log of the sum of the exponentiated values of the specified matrix of values.

Returns the log sum of exponentials.

The matrix may be a full matrix, a vector, a row vector, or a container of these.

The function is defined as follows to prevent overflow in exponential calculations.

\(\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))\).

Template Parameters

T	Type of input vector or matrix.

Parameters

[in]

x

Matrix of specified values.

Returns

The log of the sum of the exponentiated vector values.

The matrix may be a full matrix, a vector, a row vector, or a container of these.

The function is defined as follows to prevent overflow in exponential calculations.

\(\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))\).

Template Parameters

T	type of input vector or matrix

Parameters

[in]

x

matrix of specified values

Returns

The log of the sum of the exponentiated vector values.

Template Parameters

T	Type of input vector or matrix.

Parameters

x

matrix

Definition at line 55 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( double  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 57 of file log_sum_exp.hpp.

template<typename T , require_eigen_st< is_var, T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

var stan::math::log_sum_exp ( const T &  v )

inline

Returns the log sum of exponentials of the input.

Template Parameters

T	A type inheriting from EigenBase with scalar type var

Parameters

v

input

Definition at line 69 of file log_sum_exp.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

var stan::math::log_sum_exp ( const T &  x )

inline

Returns the log sum of exponentials of the input.

Template Parameters

T	A var_value with an input vector or matrix

Parameters

x

input

Definition at line 89 of file log_sum_exp.hpp.

template<typename T , typename = require_autodiff_t<T>>

double stan::math::logb

(

const T &

x

)

Returns the value of the unbiased radix-independent exponent from the floating-point argument.

Implementation note: This implementation works for all autodiff types because it is a step function, so can return a double which will produce the correct zero derivative if promoted to an autodiff variable.

Template Parameters

T	floating-point type

Parameters

[in]

x

floating-point argument

Returns

unbiased radix-independent exponent of the argument

Definition at line 24 of file logb.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_and ( const T1  x1, const T2  x2  )

inline

The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise.

Equivalent to x1 != 0 && x2 != 0.

\[ \mbox{operator\&\&}(x, y) = \begin{cases} 0 & \mbox{if } x = 0 \textrm{ or } y=0 \\ 1 & \mbox{if } x, y \neq 0 \\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true if both x1 and x2 are not equal to 0.

Definition at line 30 of file logical_and.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_eq ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is equal to the second.

Equivalent to x1 == x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true iff x1 == x2

Definition at line 19 of file logical_eq.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_gt ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is strictly greater than the second.

Equivalent to x1 < x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true if x1 > x2

Definition at line 18 of file logical_gt.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_gte ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is greater than or equal to the second.

Equivalent to x1 >= x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true if x1 >= x2

Definition at line 18 of file logical_gte.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_lt ( T1  x1, T2  x2  )

inline

Return 1 if the first argument is strictly less than the second.

Equivalent to x1 < x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true if x1 < x2

Definition at line 18 of file logical_lt.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_lte ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is less than or equal to the second.

Equivalent to x1 <= x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true iff x1 <= x2

Definition at line 18 of file logical_lte.hpp.

template<typename T >

int stan::math::logical_negation ( const T &  x )

inline

The logical negation function which returns one if the input is equal to zero and zero otherwise.

Template Parameters

T	type of value

Parameters

x

value

Returns

1 if value is zero and 0 otherwise

Definition at line 17 of file logical_negation.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_neq ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is unequal to the second.

Equivalent to x1 != x2.

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true iff x1 != x2

Definition at line 19 of file logical_neq.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_or ( T1  x1, T2  x2  )

inline

The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise.

Equivalent to x1 != 0 || x2 != 0.

\[ \mbox{operator||}(x, y) = \begin{cases} 0 & \mbox{if } x, y=0 \\ 1 & \mbox{if } x \neq 0 \textrm{ or } y\neq0\\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

x1	first argument
x2	second argument

Returns

true if either x1 or x2 is not equal to 0.

Definition at line 29 of file logical_or.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 24 of file logistic_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 24 of file logistic_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 24 of file logistic_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 27 of file logistic_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 95 of file logistic_lpdf.hpp.

template<typename T >

fvar<T> stan::math::logit ( const fvar< T > &  x )

inline

Definition at line 14 of file logit.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::logit ( const var_value< T > &  A )

inline

Returns the elementwise logit() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise logit() of the input.

Definition at line 20 of file logit.hpp.

template<typename T , require_stan_scalar_or_eigen_t< T > * = nullptr>

auto stan::math::logit ( const var_value< T > &  u )

inline

Return the log odds of the specified argument.

Template Parameters

T	Arithmetic or a type inheriting from EigenBase.

Parameters

u	The variable.

Returns

log odds of argument

Definition at line 20 of file logit.hpp.

double stan::math::logit ( double  u )

inline

Return the log odds of the argument.

The logit function is defined as for \(x \in [0, 1]\) by returning the log odds of \(x\) treated as a probability,

\(\mbox{logit}(x) = \log \left( \frac{x}{1 - x} \right)\).

The inverse to this function is inv_logit.

\[ \mbox{logit}(x) = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \ln\frac{x}{1-x} & \mbox{if } 0\leq x \leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{logit}(x)}{\partial x} = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \frac{1}{x-x^2}& \mbox{if } 0\leq x\leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log odds of argument

Definition at line 46 of file logit.hpp.

double stan::math::logit ( int  u )

inline

Return the log odds of the argument.

Parameters

u

argument

Returns

log odds of argument

Definition at line 57 of file logit.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::logit ( const Container &  x )

inline

Return the elementwise application of logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

Container

type of container

Parameters

x

container

Returns

elementwise logit of container elements

Definition at line 91 of file logit.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::logit ( const Container &  x )

inline

Version of logit() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

the logit of each variable in the container.

Note: The return must be evaluated otherwise the Ref object falls out of scope

Definition at line 108 of file logit.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::loglogistic_lpdf ( const T_y &  y, const T_scale &  alpha, const T_shape &  beta  )

inline

Definition at line 145 of file loglogistic_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 28 of file lognormal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 28 of file lognormal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 28 of file lognormal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 28 of file lognormal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 104 of file lognormal_lpdf.hpp.

template<typename T , typename L , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L, U > * = nullptr>

matrix_cl<double> stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub  )

inline

Return the lower and upper-bounded matrix derived by transforming the specified free matrix given the specified lower and upper bounds.

The transform is the transformed and scaled inverse logit,

\(f(x) = L + (U - L) \mbox{logit}^{-1}(x)\)

Template Parameters

T	matrix expression type
L	lower bound expression type
U	upper bound expression type

Parameters

[in]	x	Free matrix to transform.
[in]	lb	Lower bound
[in]	ub	Upper bound

Returns

Lower- and upper-bounded matrix derived from transforming the free matrix.

Exceptions

std::domain_error

if ub <= lb

Definition at line 35 of file lub_constrain.hpp.

template<typename T_x , typename T_lb , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb, T_ub > * = nullptr, require_any_var_t< T_x, T_lb, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb, T_ub > * = nullptr>

var_value<matrix_cl<double> > stan::math::lub_constrain ( T_x &&  x, T_lb &&  lb, T_ub &&  ub  )

inline

Return the bounded value for the specified unconstrained input and specified bounds.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T_x	type of the unconstrained input
T_lb	type of lower bound
T_ub	type of upper bound

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound
[in]	ub	upper bound

Returns

constrained matrix

Definition at line 42 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_stan_scalar_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( T &&  x, L &&  lb, U &&  ub  )

inline

Return the lower and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds.

The transform is the transformed and scaled inverse logit,

\(f(x) = L + (U - L) \mbox{logit}^{-1}(x)\)

Template Parameters

T	Scalar.
L	Scalar.
U	Scalar.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 46 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< L, U > * = nullptr>

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub, return_type_t< T, L, U > &  lp  )

inline

Return the lower and upper-bounded matrix derived by transforming the specified free matrix given the specified lower and upper bounds.

Specialization for Eigen matrix and matrix bounds plus lp.

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log density with the log absolute Jacobian determinant.

Overload for Eigen matrix and matrix bounds plus lp.

The transform is the transformed and scaled inverse logit,

\(f(x) = L + (U - L) \mbox{logit}^{-1}(x)\)

Template Parameters

T	matrix expression type
L	lower bound expression type
U	upper bound expression type

Parameters

[in]	x	Free matrix to transform.
[in]	lb	Lower bound
[in]	ub	Upper bound
[in,out]	lp	Log probability scalar reference

Returns

Lower- and upper-bounded matrix derived from transforming the free matrix.

Exceptions

std::domain_error

if ub <= lb

The transform is as defined in lub_constrain(T, double, double). The log absolute Jacobian determinant is given by

\(\log \left| \frac{d}{dx} \left( L + (U-L) \mbox{logit}^{-1}(x) \right) \right|\)

\( {} = \log | (U-L) \, (\mbox{logit}^{-1}(x)) \, (1 - \mbox{logit}^{-1}(x)) |\)

\( {} = \log (U - L) + \log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))\)

Template Parameters

T	Scalar.
L	Scalar.
U	Scalar.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.
[in,out]	lp	Log probability scalar reference.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 74 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_stan_scalar_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( T &&  x, L &&  lb, U &&  ub, return_type_t< T, L, U > &  lp  )

inline

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log density with the log absolute Jacobian determinant.

The transform is as defined in lub_constrain(T, double, double). The log absolute Jacobian determinant is given by

\(\log \left| \frac{d}{dx} \left( L + (U-L) \mbox{logit}^{-1}(x) \right) \right|\)

\( {} = \log | (U-L) \, (\mbox{logit}^{-1}(x)) \, (1 - \mbox{logit}^{-1}(x)) |\)

\( {} = \log (U - L) + \log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))\)

Template Parameters

T	Scalar.
L	Scalar.
U	Scalar.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.
[in,out]	lp	Log probability scalar reference.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 97 of file lub_constrain.hpp.

template<typename T_x , typename T_lb , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_lb, T_ub > * = nullptr, require_any_var_t< T_x, T_lb, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_lb, T_ub > * = nullptr>

var_value<matrix_cl<double> > stan::math::lub_constrain ( T_x &&  x, T_lb &&  lb, T_ub &&  ub, var &  lp  )

inline

Return the bounded value for the specified unconstrained input and specified bounds.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

Template Parameters

T_x	type of the unconstrained input
T_lb	type of lower bound
T_ub	type of upper bound

Parameters

[in]	x	unconstrained input
[in]	lb	lower bound
[in]	ub	upper bound
[in,out]	lp	reference to log probability to increment

Returns

constrained matrix

Definition at line 98 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_eigen_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub  )

inline

Overload for Eigen matrix and scalar bounds.

Specialization for Eigen matrix with scalar lower bound and matrix upper bound.

Specialization for Eigen matrix with matrix lower bound and scalar upper bound.

Specialization for Eigen matrix and scalar bounds.

Overload for Eigen matrix with scalar lower bound and matrix upper bound.

Overload for Eigen matrix with matrix lower bound and scalar upper bound.

Definition at line 120 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_eigen_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub, return_type_t< T, L, U > &  lp  )

inline

Overload for Eigen matrix and scalar bounds plus lp.

Specialization for Eigen matrix and scalar bounds plus lp.

Overload for Eigen matrix with scalar lower bound and matrix upper bound plus lp.

Overload for Eigen matrix with matrix lower bound and scalar upper bound plus lp.

Definition at line 131 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_eigen_t< T, L, U > * = nullptr, require_not_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub  )

inline

Overload for Eigen matrix and matrix bounds.

Specialization for Eigen matrix and matrix bounds.

Return the lower and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds.

The transform is the transformed and scaled inverse logit,

\(f(x) = L + (U - L) \mbox{logit}^{-1}(x)\)

Template Parameters

T	Scalar.
L	Scalar.
U	Scalar.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 203 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const L &  lb, const U &  ub  )

inline

Overload for array of x and non-array lb and ub.

Definition at line 247 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const L &  lb, const U &  ub, return_type_t< T, L, U > &  lp  )

inline

Overload for array of x and non-array lb and ub with lp.

Definition at line 261 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const L &  lb, const std::vector< U > &  ub  )

inline

Overload for array of x and ub and non-array lb.

Definition at line 276 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const L &  lb, const std::vector< U > &  ub, return_type_t< T, L, U > &  lp  )

inline

Overload for array of x and ub and non-array lb with lp.

Definition at line 292 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_all_matrix_t< T, L > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_var_t< return_type_t< T, L, U >> * = nullptr>

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub, std::decay_t< return_type_t< T, L, U >> &  lp  )

inline

Specialization for Eigen matrix with matrix lower bound and scalar upper bound plus lp.

Specialization for Eigen matrix with scalar lower bound and matrix upper bound plus lp.

Definition at line 305 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb, const U &  ub  )

inline

Overload for array of x and lb and non-array ub.

Definition at line 309 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::lub_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb, const U &  ub, return_type_t< T, L, U > &  lp  )

inline

Overload for array of x and lb and non-array ub with lp.

Definition at line 325 of file lub_constrain.hpp.

template<typename T , typename L , typename U >

auto stan::math::lub_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb, const std::vector< U > &  ub  )

inline

Overload for array of x, lb, and ub with lp.

Definition at line 340 of file lub_constrain.hpp.

template<typename T , typename L , typename U >

auto stan::math::lub_constrain ( const std::vector< T > &  x, const std::vector< L > &  lb, const std::vector< U > &  ub, return_type_t< T, L, U > &  lp  )

inline

Overload for array of x, lb, and ub.

Definition at line 356 of file lub_constrain.hpp.

template<bool Jacobian, typename T , typename L , typename U >

auto stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub, return_type_t< T, L, U > &  lp  )

inline

Return the lower and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
L	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
U	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

[in]	x	Free scalar to transform
[in]	lb	Lower bound
[in]	ub	Upper bound
[in,out]	lp	log density accumulator

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar

Exceptions

std::domain_error

if ub <= lb

Definition at line 394 of file lub_constrain.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< T > * = nullptr, require_all_stan_scalar_t< L, U > * = nullptr>

auto stan::math::lub_free ( T &&  y, L &&  lb, U &&  ub  )

inline

lub_free Return the unconstrained variable that transforms to the y given the specified bounds.

Overload for matrix constrained variable, matrix upper bound, scalar lower bound.

Overload for matrix constrained variable, matrix lower bound, scalar upper bound.

The transform in lub_constrain, is reversed by a transformed and scaled logit,

\(f^{-1}(y) = \mbox{logit}(\frac{y - L}{U - L})\)

where \(U\) and \(L\) are the lower and upper bounds.

Template Parameters

T	type of bounded object
L	type of lower bound
U	type of upper bound

Parameters

y	constrained value
lb	lower bound
ub	upper bound

Returns

the free object that transforms to the input given the bounds

Exceptions

std::invalid_argument

if the lower bound is greater than the upper bound, y is less than the lower bound, or y is greater than the upper bound Overload for all scalar arguments

Definition at line 46 of file lub_free.hpp.

template<typename T , typename L , typename U , require_all_eigen_t< T, L, U > * = nullptr>

auto stan::math::lub_free ( T &&  y, L &&  lb, U &&  ub  )

inline

Overload for matrix constrained variable, matrix upper bound, matrix lower bound.

Definition at line 110 of file lub_free.hpp.

template<typename T , typename L , typename U , require_all_not_std_vector_t< L, U > * = nullptr>

auto stan::math::lub_free ( const std::vector< T >  y, const L &  lb, const U &  ub  )

inline

Overload for std::vector constrained variable.

Definition at line 131 of file lub_free.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< L > * = nullptr>

auto stan::math::lub_free ( const std::vector< T >  y, const L &  lb, const std::vector< U > &  ub  )

inline

Overload for std::vector constrained variable and std::vector upper bound.

Definition at line 144 of file lub_free.hpp.

template<typename T , typename L , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::lub_free ( const std::vector< T >  y, const std::vector< L > &  lb, const U &  ub  )

inline

Overload for std::vector constrained variable and std::vector lower bound.

Definition at line 159 of file lub_free.hpp.

template<typename T , typename L , typename U >

auto stan::math::lub_free ( const std::vector< T >  y, const std::vector< L > &  lb, const std::vector< U > &  ub  )

inline

Overload for std::vector constrained variable and std::vector constraints.

Definition at line 173 of file lub_free.hpp.

static constexpr double stan::math::machine_precision ( )

inlinestatic

Returns the difference between 1.0 and the next value representable.

Returns

Minimum positive number.

Definition at line 221 of file constants.hpp.

template<typename T , typename F >

var_value<plain_type_t<T> > stan::math::make_callback_var	(	T &&	value,
		F &&	functor
	)

Creates a new var initialized with a callback_vari with a given value and reverse-pass callback functor.

The callback functor will be passed a reference to the constructed vari.

All captured values must be trivially destructible or they will leak memory. to_arena() function can be used to ensure that.

Template Parameters

T	type of value
F	type of callable

Parameters

value	value of the vari
functor	functor or other callable to call in the reverse pass

Definition at line 61 of file callback_vari.hpp.

template<typename T , typename F >

internal::callback_vari<plain_type_t<T>, F>* stan::math::make_callback_vari	(	T &&	value,
		F &&	functor
	)

Creates a new vari with given value and a callback that implements the reverse pass (chain).

The callback needs to accept a referenct to the vari. If it needs any other data it should be implemented as a lambda capturing the variables it needs.

All captured values must be trivially destructible or they will leak memory! to_arena() function can be used to ensure that.

Template Parameters

T	type of value
F	type of callable

Parameters

value	value of the vari
functor	functor or other callable to call in the reverse pass

Definition at line 41 of file callback_vari.hpp.

template<typename T >

auto stan::math::make_chainable_ptr

(

T &&

obj

)

Store the given object in a chainable_object so it is destructed only when the chainable stack memory is recovered and return a pointer to the underlying object.

Template Parameters

T	type of object to hold

Parameters

obj	object to hold

Returns

pointer to object held in chainable_object

Definition at line 60 of file chainable_object.hpp.

template<typename F , typename... Args, require_not_plain_type_t< decltype(std::declval< F >()(std::declval< Args & >()...))> * = nullptr>

auto stan::math::make_holder	(	const F &	func,
		Args &&...	args
	)

Constructs an expression from given arguments using given functor.

Calls given function with given arguments.

This is similar to calling the functor with given arguments. Except that any rvalue argument will be moved to heap first. The arguments moved to heap are deleted once the expression is destructed.

Template Parameters

F	type of the functor
Args	types of the arguments

Parameters

func	the functor
args	arguments for the functor

Returns

holder referencing expression constructed by given functor

No holder is necessary if the function is not returning Eigen expression.

Template Parameters

F	type of the functor
Args	types of the arguments

Parameters

func	the functor
args	arguments for the functor

Returns

holder referencing expression constructed by given functor

Definition at line 352 of file holder.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::make_ldlt_factor ( const T &  A )

inline

Make an LDLT_factor with matrix type plain_type_t<T>

Template Parameters

T	Type of matrix to take the LDLT of

Parameters

A	Matrix to take the LDLT of

Returns

LDLT_factor of A

Definition at line 62 of file LDLT_factor.hpp.

template<typename T >

Eigen::Array<T, Eigen::Dynamic, 1> stan::math::make_nu	(	const T &	eta,
		size_t	K
	)

Return the degrees of freedom for the t distribution that corresponds to the shape parameter in the LKJ distribution.

Template Parameters

T	scalar type

Parameters

eta	LKJ distribution parameter in (0, inf)
K	number of variables in correlation matrix

Definition at line 19 of file make_nu.hpp.

template<typename T >

auto stan::math::make_unsafe_chainable_ptr

(

T &&

obj

)

Store the given object in a chainable_object so it is destructed only when the chainable stack memory is recovered and return a pointer to the underlying object This function differs from make_chainable_object in that this class does not evaluate expressions.

Template Parameters

T	type of object to hold

Parameters

obj	object to hold

Returns

pointer to object held in chainable_object

Definition at line 115 of file chainable_object.hpp.

template<int call_id, typename F , typename T_shared_param , typename T_job_param , require_eigen_col_vector_t< T_shared_param > * = nullptr>

Eigen::Matrix<return_type_t<T_shared_param, T_job_param>, Eigen::Dynamic, 1> stan::math::map_rect	(	const T_shared_param &	shared_params,
		const std::vector< Eigen::Matrix< T_job_param, Eigen::Dynamic, 1 >> &	job_params,
		const std::vector< std::vector< double >> &	x_r,
		const std::vector< std::vector< int >> &	x_i,
		std::ostream *	msgs = nullptr
	)

Map N function evaluations to parameters and data which are in rectangular format.

Each function evaluation may return a column vector of different sizes and the output is the concatenated vector from all evaluations.

In addition to job specific parameters, real and int data, a shared parameter vector is repeated in all evaluations. All input parameters are stored as vectors whereas data is stored as arrays.

For N jobs the output of this function is

[ f(shared_params, job_params[1], x_r[1], x_i[1]), f(shared_params, job_params[2], x_r[2], x_i[2]), ... ]'.

The function is implemented with serial execution and with parallelism using threading or MPI (TODO). The threading version is used if the compiler flag STAN_THREADS is set during compilation while the MPI version is only available if STAN_MPI is defined. The MPI parallelism takes precedence over serial or threading execution of the function.

For the threaded parallelism the N jobs are chunked into T blocks which are executed asynchronously using the async C++11 facility. This ensure that at most T threads are used, but the actual number of threads is controlled by the implementation of async provided by the compiler. Note that nested calls of map_rect will lead to a multiplicative increase in the number of job chunks generated. The number of threads T is controlled at runtime via the STAN_NUM_threads environment variable, see the get_num_threads function for details.

For the MPI version to work this function has these special non-standard conventions:

• The call_id template parameter is considered as a label for the functor F and data combination. Since MPI communication is expensive, the real and int data is transmitted only a single time per functor F / call_id combination to the workers.
• The MPI implementation requires that the functor type fully specifies the functor and hence requires a default constructible function object. This choice reduces the need for communication across MPI as the type is sufficient and the state of the functor is not necessary to transmit. Thus, the functor is specified as template argument only.
• The size of the returned vector of each job must stay consistent when performing repeated calls to map_rect (which usually vary the values of the parameters).
• To achieve the exact same results between the serial and the MPI evaluation scheme both variants work in exactly the same way which is to use nested AD calls and pre-computing all gradients when the function gets called. The usual evaluation scheme would build an AD graph instead of doing on-the-spot gradient evaluation. For large problems this results in speedups for the serial version even on a single core due to smaller AD graph sizes.
• In MPI operation mode, no outputs from the workers are streamed back to the root.
• Finally, each map_rect call must be registered with the STAN_REGISTER_MAP_RECT macro. This is required to enable communication between processes for the MPI case. The macro definition is empty if MPI is not enabled.

The functor F is expected to have the usual operator() function with signature

template <typename T1, typename T2> Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> operator()(const Eigen::Matrix<T1, Eigen::Dynamic, 1>& eta, const Eigen::Matrix<T2, Eigen::Dynamic, 1>& theta, const std::vector<double>& x_r, const std::vector<int>& x_i, std::ostream* msgs = 0) const { ... }

If an expression is passed as shared_params, the functor needs to accept expression.

WARNING: For the MPI case, the data arguments are NOT checked if they are unchanged between repeated evaluations for a given call_id/functor F pair. This is silently assumed to be immutable between evaluations.

Template Parameters

T_shared_param	Type of shared parameters.
T_job_param	Type of job specific parameters.

Parameters

shared_params	shared parameter vector passed as first argument to functor for all jobs
job_params	Array of job specific parameter vectors. All job specific parameters must have matching sizes.
x_r	Array of real arrays for each job. The first dimension must match the number of jobs (which is determined by the first dimension of job_params) and each entry must have the same size.
x_i	Array of int data with the same conventions as x_r.
msgs	Output stream for messages.

Template Parameters

call_id	Label for functor/data combination. See above for details.
F	Functor which is applied to all job specific parameters with conventions described.

Returns

concatenated results from all jobs

Definition at line 127 of file map_rect.hpp.

template<typename T , typename = require_eigen_t<T>>

plain_type_t<T> stan::math::matrix_exp ( const T &  A_in )

inline

Return the matrix exponential of the input matrix.

Template Parameters

T	type of the matrix

Parameters

[in]

A_in

Matrix to exponentiate.

Returns

Matrix exponential, dynamically-sized.

Exceptions

<code>std::invalid_argument</code>

if the input matrix is not square.

Definition at line 25 of file matrix_exp.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_exp_2x2

(

const EigMat &

A

)

Return the matrix exponential of a 2x2 matrix.

Reference for algorithm: http://mathworld.wolfram.com/MatrixExponential.html Note: algorithm only works if delta > 0;

Template Parameters

EigMat

type of the matrix

Parameters

[in]

A

2x2 matrix to exponentiate.

Returns

Matrix exponential of A.

Definition at line 26 of file matrix_exp_2x2.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_st_same< double, EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<double, Eigen::Dynamic, EigMat2::ColsAtCompileTime> stan::math::matrix_exp_multiply ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return product of exp(A) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double.

Template Parameters

Cb	number of columns in matrix B, can be Eigen::Dynamic

Parameters

[in]	A	Matrix
[in]	B	Matrix

Returns

exponential of A multiplies B

Definition at line 24 of file matrix_exp_multiply.hpp.

template<typename Ta , typename Tb , require_all_eigen_t< Ta, Tb > * = nullptr, require_any_st_autodiff< Ta, Tb > * = nullptr>

Eigen::Matrix<return_type_t<Ta, Tb>, -1, Tb::ColsAtCompileTime> stan::math::matrix_exp_multiply ( const Ta &  A, const Tb &  B  )

inline

Wrapper of matrix_exp_action function for a more literal name.

Template Parameters

Ta	type of the matrix A
Tb	type of the matrix B

Parameters

[in]	A	Matrix
[in]	B	Matrix

Returns

exponential of A multiplies B

Definition at line 27 of file matrix_exp_multiply.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime> stan::math::matrix_exp_pade

(

const EigMat &

arg

)

Computes the matrix exponential, using a Pade approximation, coupled with scaling and squaring.

Template Parameters

MatrixType

type of the matrix

Parameters

[in]

arg

matrix to exponentiate.

Returns

Matrix exponential of input matrix.

Definition at line 23 of file matrix_exp_pade.hpp.

template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D , require_all_matrix_t< T_y, T_Mu, T_Sigma, T_D > * = nullptr>

return_type_t<T_y, T_Mu, T_Sigma, T_D> stan::math::matrix_normal_prec_lpdf	(	const T_y &	y,
		const T_Mu &	Mu,
		const T_Sigma &	Sigma,
		const T_D &	D
	)

Definition at line 84 of file matrix_normal_prec_lpdf.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime> stan::math::matrix_power ( const EigMat &  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

T	type of the matrix

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 26 of file matrix_power.hpp.

template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>

plain_type_t<T_m> stan::math::matrix_power ( T_m &&  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

T	type of the matrix

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 27 of file matrix_power.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

plain_type_t<T> stan::math::matrix_power ( const T &  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 29 of file matrix_power.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::matrix_power ( const var_value< T > &  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

T	type of the matrix

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 29 of file matrix_power.hpp.

template<typename T_matrix , typename T_vector , typename = require_all_kernel_expressions_t<T_matrix, T_vector>>

auto stan::math::matrix_vector_multiply ( T_matrix &&  matrix, T_vector &&  vector  )

inline

Multiplies a matrix and a vector on an OpenCL device.

Template Parameters

T_matrix	type of the matrix or matrix expression
T_vector	type of the vector or vector expression

Parameters

matrix	the matrix
vector	the vector

Returns

matrix-vector multiplication expression

Exceptions

invalid_argument

vector has more than column or matrix has different number of rows than vector columns.

Definition at line 27 of file matrix_vector_multiply.hpp.

template<typename T , require_container_t< T > * = nullptr>

value_type_t<T> stan::math::max ( const T &  m )

inline

Returns the maximum coefficient in the specified matrix, vector, row vector or std vector.

Template Parameters

T	type of the container

Parameters

m	specified matrix, vector, row vector or std vector

Returns

maximum coefficient value in the container, or -infinity if the container is size zero and the scalar type in container is floating point number

Exceptions

<code>std::invalid_argument</code>

if the vector is size zero and the scalar type in the container is integer

Definition at line 27 of file max.hpp.

template<typename T1 , typename... Ts>

size_t stan::math::max_size ( const T1 &  x1, const Ts &...  xs  )

inline

Calculate the size of the largest input.

Template Parameters

T1	type of the first input
Ts	types of the other inputs

Parameters

x1	first input
xs	other inputs

Returns

the size of the largest input

Definition at line 19 of file max_size.hpp.

template<typename T1 , typename... Ts>

size_t stan::math::max_size_mvt ( const T1 &  x1, const Ts &...  xs  )

inline

Calculate the size of the largest multivariate input.

A multivariate container is either an Eigen matrix, whose mvt size is 1, or an std::vector of Eigen matrices, whose mvt size is the size of the std::vector. It is an error to supply any other type of input.

Template Parameters

T1	type of the first input
Ts	types of the other inputs

Parameters

x1	first input
xs	other inputs

Returns

the size of the largest input

Exceptions

<tt>invalid_argument</tt>

if provided with an input that is not an Eigen matrix or std::vector of Eigen matrices

Definition at line 24 of file max_size_mvt.hpp.

template<typename T1 , typename T2 , require_all_eigen_vt< is_fvar, T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>

Eigen::Matrix< value_type_t< T1 >, T1::RowsAtCompileTime, T2::ColsAtCompileTime > stan::math::mdivide_left ( const T1 &  A, const T2 &  b  )

inline

Definition at line 21 of file mdivide_left.hpp.

template<typename T1 , typename T2 , require_all_eigen_vt< std::is_arithmetic, T1, T2 > * = nullptr>

Eigen::Matrix<return_type_t<T1, T2>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left ( const T1 &  A, const T2 &  b  )

inline

Returns the solution of the system Ax=b.

Template Parameters

T1	type of the first matrix
T2	type of the right-hand side matrix or vector

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 27 of file mdivide_left.hpp.

template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>

auto stan::math::mdivide_left ( const T1 &  A, const T2 &  B  )

inline

Return the solution X of AX = B.

A must be a square matrix, but B can be a matrix or a vector

Template Parameters

T1	type of first matrix
T2	type of second matrix

Parameters

[in]	A	square matrix
[in]	B	right hand side

Returns

solution of AX = B

Definition at line 30 of file mdivide_left.hpp.

template<typename T1 , typename T2 , require_eigen_vt< std::is_arithmetic, T1 > * = nullptr, require_eigen_vt< is_fvar, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T2>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left ( const T1 &  A, const T2 &  b  )

inline

Definition at line 62 of file mdivide_left.hpp.

template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>

auto stan::math::mdivide_left_ldlt ( LDLT_factor< T1 > &  A, const T2 &  B  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Template Parameters

T

type of B

Parameters

A	LDLT_factor
B	Right hand side matrix or vector.

Returns

x = A^-1 B, solution of the linear system.

Exceptions

std::domain_error

if rows of B don't match the size of A.

Definition at line 27 of file mdivide_left_ldlt.hpp.

template<typename T , typename EigMat , require_eigen_vt< std::is_arithmetic, T > * = nullptr, require_eigen_vt< is_fvar, EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, EigMat::ColsAtCompileTime> stan::math::mdivide_left_ldlt ( LDLT_factor< T > &  A, const EigMat &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Template Parameters

T	type of matrix for the LDLT_factor
EigMat	type of the right-hand side matrix or vector

Parameters

A	LDLT_factor
b	right-hand side matrix or vector

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 28 of file mdivide_left_ldlt.hpp.

template<typename T , typename EigMat , require_eigen_t< EigMat > * = nullptr, require_all_not_st_var< T, EigMat > * = nullptr, require_any_not_t< std::is_arithmetic< value_type_t< T >>, is_fvar< value_type_t< EigMat >>> * = nullptr>

Eigen::Matrix<return_type_t<T, EigMat>, Eigen::Dynamic, EigMat::ColsAtCompileTime> stan::math::mdivide_left_ldlt ( LDLT_factor< T > &  A, const EigMat &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Template Parameters

T	type of matrix in the LDLT_factor
EigMat	type of the right hand side

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 30 of file mdivide_left_ldlt.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_var< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<return_type_t<EigMat1, EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_left_spd ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Returns the solution of the system Ax=b where A is symmetric positive definite.

Template Parameters

EigMat1	type of the first matrix
EigMat2	type of the right-hand side matrix or vector

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_left_spd.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_matrix_base_vt< is_var, EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<var, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_left_spd ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 152 of file mdivide_left_spd.hpp.

template<typename EigMat1 , typename EigMat2 , require_eigen_matrix_base_vt< is_var, EigMat1 > * = nullptr, require_eigen_matrix_base_vt< std::is_arithmetic, EigMat2 > * = nullptr>

Eigen::Matrix< var, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime > stan::math::mdivide_left_spd ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 183 of file mdivide_left_spd.hpp.

template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::mdivide_left_spd ( const T1 &  A, const T2 &  B  )

inline

Returns the solution of the system Ax=B where A is symmetric positive definite.

This overload handles arguments where one of T1 or T2 are var_value<T> where T is an Eigen type. The other type can also be a var_value or it can be a matrix type that inherits from EigenBase

Template Parameters

T1	type of the first matrix
T2	type of the right-hand side matrix or vector

Parameters

A	Matrix.
B	Right hand side matrix or vector.

Returns

x = A^-1 B, solution of the linear system.

Exceptions

std::domain_error

if A is not square or B does not have as many rows as A has columns.

Definition at line 261 of file mdivide_left_spd.hpp.

template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_all_not_eigen_vt< is_var, T1, T2 > * = nullptr>

Eigen::Matrix<return_type_t<T1, T2>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri ( const T1 &  A, const T2 &  b  )

inline

Returns the solution of the system Ax=b when A is triangular.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
T1	type of the triangular matrix
T2	type of the right-hand side matrix or vector

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 31 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, typename T , require_eigen_t< T > * = nullptr>

plain_type_t<T> stan::math::mdivide_left_tri ( const T &  A )

inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Template Parameters

T	type of the matrix

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 55 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_eigen_vt< is_var, T1, T2 > * = nullptr>

Eigen::Matrix<var, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri ( const T1 &  A, const T2 &  b  )

inline

Definition at line 248 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_eigen_vt< std::is_arithmetic, T1 > * = nullptr, require_eigen_vt< is_var, T2 > * = nullptr>

Eigen::Matrix< var, T1::RowsAtCompileTime, T2::ColsAtCompileTime > stan::math::mdivide_left_tri ( const T1 &  A, const T2 &  b  )

inline

Definition at line 274 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::mdivide_left_tri ( const T1 &  A, const T2 &  B  )

inline

Returns the solution of the system Ax=B when A is triangular.

This overload handles arguments where one of T1 or T2 are var_value<T> where T is an Eigen type. The other type can also be a var_value or it can be a matrix type that inherits from EigenBase

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
T1	type of the triangular matrix
T2	type of the right-hand side matrix or vector

Parameters

A	Triangular matrix.
B	Right hand side matrix or vector.

Returns

x = A^-1 B, solution of the linear system.

Exceptions

std::domain_error

if A is not square or B does not have as many rows as A has columns.

Definition at line 345 of file mdivide_left_tri.hpp.

template<typename T1 , typename T2 , require_all_eigen_vt< is_fvar, T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T1>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri_low ( const T1 &  A, const T2 &  b  )

inline

Definition at line 20 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr>

matrix_cl<double> stan::math::mdivide_left_tri_low ( const T1 &  A, const T2 &  b  )

inline

Returns the solution of the system Ax=b when A is lower triangular.

Template Parameters

T1	type of elements in A
T2	type of elements in b

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 26 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::mdivide_left_tri_low ( T1 &&  A, T2 &&  b  )

inline

Returns the solution of the system Ax=b when A is lower triangular.

Template Parameters

T1	type of elements in A
T2	type of elements in b

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_all_not_eigen_vt< is_fvar, T1, T2 > * = nullptr>

Eigen::Matrix<return_type_t<T1, T2>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri_low ( const T1 &  A, const T2 &  b  )

inline

Return the result of left dividing the second argument by the first argument.

Calling mdivide_left_tri_low(A, b) with divisor A and dividend b is more arithmetically stable than calling inv(A) * b.

Template Parameters

T1	type of the divisor matrix
T2	type of the dividend matrix

Parameters

A	divisor, an invertible square matrix
b	dividend, a matrix or vector with the same number of rows as the divisor has columns

Returns

left division of b by A

Exceptions

std::invalid_argument

if the divisor is not square or the dividend does not have the same number of rows as the divisor has columns.

Definition at line 33 of file mdivide_left_tri_low.hpp.

template<typename T , require_all_kernel_expressions_t< T > * = nullptr>

matrix_cl<double> stan::math::mdivide_left_tri_low ( const T &  A )

inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Template Parameters

T	type of elements in A

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 44 of file mdivide_left_tri_low.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_fvar, T > * = nullptr>

plain_type_t<T> stan::math::mdivide_left_tri_low ( const T &  A )

inline

Definition at line 45 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , require_eigen_t< T1 > * = nullptr, require_vt_same< double, T1 > * = nullptr, require_eigen_vt< is_fvar, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T2>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri_low ( const T1 &  A, const T2 &  b  )

inline

Definition at line 57 of file mdivide_left_tri_low.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::mdivide_left_tri_low ( const var_value< T > &  A )

inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Template Parameters

T	type of elements in A

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 65 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , require_eigen_vt< is_fvar, T1 > * = nullptr, require_eigen_t< T2 > * = nullptr, require_vt_same< double, T2 > * = nullptr>

Eigen::Matrix<value_type_t<T1>, T1::RowsAtCompileTime, T2::ColsAtCompileTime> stan::math::mdivide_left_tri_low ( const T1 &  A, const T2 &  b  )

inline

Definition at line 87 of file mdivide_left_tri_low.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< is_fvar, EigMat1, EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix< value_type_t< EigMat1 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime > stan::math::mdivide_right ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 21 of file mdivide_right.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_fvar< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<return_type_t<EigMat1, EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right ( const EigMat1 &  b, const EigMat2 &  A  )

inline

Returns the solution of the system xA=b.

Template Parameters

EigMat1	type of the right-hand side matrix or vector
EigMat2	type of the second matrix

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 28 of file mdivide_right.hpp.

template<typename EigMat1 , typename EigMat2 , require_eigen_vt< std::is_arithmetic, EigMat1 > * = nullptr, require_eigen_vt< is_fvar, EigMat2 > * = nullptr>

Eigen::Matrix<value_type_t<EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 97 of file mdivide_right.hpp.

template<typename EigMat , typename T , require_all_matrix_t< EigMat, T > * = nullptr, require_any_not_st_arithmetic< EigMat, T > * = nullptr>

auto stan::math::mdivide_right_ldlt ( const EigMat &  b, LDLT_factor< T > &  A  )

inline

Returns the solution of the system xA=b given an LDLT_factor of A.

Template Parameters

EigMat	type of the right hand side
T	type of matrix in the LDLT_factor

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 27 of file mdivide_right_ldlt.hpp.

template<typename EigMat , typename T , require_all_matrix_t< EigMat, T > * = nullptr, require_all_st_arithmetic< EigMat, T > * = nullptr>

Eigen::Matrix<double, EigMat::RowsAtCompileTime, T::ColsAtCompileTime> stan::math::mdivide_right_ldlt ( const EigMat &  b, LDLT_factor< T > &  A  )

inline

Returns the solution of the system xA=b given an LDLT_factor of A.

Overload for arithmetic types

Template Parameters

EigMat	type of the right hand side
T	type of matrix in the LDLT_factor

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 51 of file mdivide_right_ldlt.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<return_type_t<EigMat1, EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right_spd ( const EigMat1 &  b, const EigMat2 &  A  )

inline

Returns the solution of the system xA=b where A is symmetric positive definite.

Template Parameters

EigMat1	type of the right-hand side matrix or vector
EigMat2	type of the second matrix

Parameters

b	right-hand side matrix or vector
A	matrix

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_right_spd.hpp.

template<Eigen::UpLoType TriView, typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<return_type_t<EigMat1, EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right_tri ( const EigMat1 &  b, const EigMat2 &  A  )

inline

Returns the solution of the system xA=b when A is triangular.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
EigMat1	type of the right-hand side matrix or vector
EigMat2	type of the triangular matrix

Parameters

A	Triangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 31 of file mdivide_right_tri.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< is_fvar, EigMat1, EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix< value_type_t< EigMat1 >, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime > stan::math::mdivide_right_tri_low ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 19 of file mdivide_right_tri_low.hpp.

template<typename T1 , typename T2 , require_all_kernel_expressions_t< T1, T2 > * = nullptr>

matrix_cl<double> stan::math::mdivide_right_tri_low ( const T2 &  b, const T1 &  A  )

inline

Returns the solution of the system Ax=b where A is a lower triangular matrix.

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = b * tri(A)^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 25 of file mdivide_right_tri_low.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_all_not_vt_fvar< EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<return_type_t<EigMat1, EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right_tri_low ( const EigMat1 &  b, const EigMat2 &  A  )

inline

Returns the solution of the system x tri(A) = b when tri(A) is a lower triangular view of the matrix A.

Template Parameters

EigMat1	type of the right-hand side matrix or vector
EigMat2	type of the second matrix

Parameters

b	right-hand side matrix or vector
A	matrix

Returns

x = b * tri(A)^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 29 of file mdivide_right_tri_low.hpp.

template<typename T1 , typename T2 , require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::mdivide_right_tri_low ( T2 &&  b, T1 &&  A  )

inline

Returns the solution of the system Ax=b when A is lower triangular.

Template Parameters

T1	type of elements in A
T2	type of elements in b

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_right_tri_low.hpp.

template<typename EigMat1 , typename EigMat2 , require_eigen_vt< std::is_arithmetic, EigMat1 > * = nullptr, require_eigen_vt< is_fvar, EigMat2 > * = nullptr>

Eigen::Matrix<value_type_t<EigMat2>, EigMat1::RowsAtCompileTime, EigMat2::ColsAtCompileTime> stan::math::mdivide_right_tri_low ( const EigMat1 &  A, const EigMat2 &  b  )

inline

Definition at line 98 of file mdivide_right_tri_low.hpp.

template<typename T , require_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

scalar_type_t<T> stan::math::mean

(

const T &

m

)

Returns the sample mean (i.e., average) of the coefficients in the specified std vector, vector, row vector, or matrix.

Parameters

m	input kernel generator expression

Returns

Sample mean of the input kernel expressions.

Definition at line 20 of file mean.hpp.

template<typename T , require_container_t< T > * = nullptr>

return_type_t<T> stan::math::mean ( const T &  m )

inline

Returns the sample mean (i.e., average) of the coefficients in the specified std vector, vector, row vector, or matrix.

Template Parameters

T	type of the matrix

Parameters

m	Specified std vector, vector, row vector, or matrix.

Returns

Sample mean of container coefficients.

Definition at line 22 of file mean.hpp.

template<typename T , require_container_t< T > * = nullptr>

value_type_t<T> stan::math::min ( const T &  m )

inline

Returns the minimum coefficient in the specified matrix, vector, row vector or std vector.

Template Parameters

T	type of elements in the container

Parameters

m	specified matrix, vector, row vector or std vector

Returns

mainimum coefficient value in the container, or infinity if the container is size zero and the scalar type in container is floating point number

Exceptions

<code>std::invalid_argument</code>

if the vector is size zero and the scalar type in the container is integer

Definition at line 26 of file min.hpp.

template<typename T >

auto stan::math::minus ( const T &  x )

inline

Returns the negation of the specified scalar or matrix.

Template Parameters

T	Type of subtrahend.

Parameters

x	Subtrahend.

Returns

Negation of subtrahend.

Definition at line 17 of file minus.hpp.

template<typename T >

fvar<T> stan::math::modified_bessel_first_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file modified_bessel_first_kind.hpp.

var stan::math::modified_bessel_first_kind ( int  v, const var &  a  )

inline

Definition at line 26 of file modified_bessel_first_kind.hpp.

template<typename T2 , require_arithmetic_t< T2 > * = nullptr>

T2 stan::math::modified_bessel_first_kind ( int  v, const T2  z  )

inline

\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_first\_kind}(v, z)}{\partial z} = \begin{cases} \frac{\partial\, I_v(z)}{\partial z} & \mbox{if } -\infty\leq z\leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {I_v}(z) = \left(\frac{1}{2}z\right)^v\sum_{k=0}^\infty \frac{\left(\frac{1}{4}z^2\right)^k}{k!\Gamma(v+k+1)} \]

\[ \frac{\partial \, I_v(z)}{\partial z} = I_{v-1}(z)-\frac{v}{z}I_v(z) \]

Definition at line 38 of file modified_bessel_first_kind.hpp.

double stan::math::modified_bessel_first_kind ( int  v, int  z  )

inline

This function exists because when z is of type integer, cyl_bessel_i(v, z) returns an integer.

This results in overflow when the function value is large.

Definition at line 49 of file modified_bessel_first_kind.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::modified_bessel_first_kind ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the modified_bessel_first_kind function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

modified_bessel_first_kind function applied to the two inputs.

Definition at line 66 of file modified_bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::modified_bessel_second_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file modified_bessel_second_kind.hpp.

var stan::math::modified_bessel_second_kind ( int  v, const var &  a  )

inline

Definition at line 26 of file modified_bessel_second_kind.hpp.

template<typename T2 , require_arithmetic_t< T2 > * = nullptr>

T2 stan::math::modified_bessel_second_kind ( int  v, const T2  z  )

inline

\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_second\_kind}(v, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ \frac{\partial\, K_v(z)}{\partial z} & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {K_v}(z) = \frac{\pi}{2}\cdot\frac{I_{-v}(z) - I_{v}(z)}{\sin(v\pi)} \]

\[ \frac{\partial \, K_v(z)}{\partial z} = -\frac{v}{z}K_v(z)-K_{v-1}(z) \]

Definition at line 43 of file modified_bessel_second_kind.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::modified_bessel_second_kind ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the modified_bessel_second_kind function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

modified_bessel_second_kind function applied to the two inputs.

Definition at line 58 of file modified_bessel_second_kind.hpp.

int stan::math::modulus ( int  x, int  y  )

inline

Definition at line 12 of file modulus.hpp.

template<typename T >

std::unique_lock< std::mutex > stan::math::mpi_broadcast_command

(

)

Broadcasts default constructible commands to the cluster.

Template Parameters

T	default constructible command class derived from mpi_command

Returns

A unique_lock instance locking the mpi_cluster

Definition at line 238 of file mpi_cluster.hpp.

std::unique_lock<std::mutex> stan::math::mpi_broadcast_command ( std::shared_ptr< mpi_command > &  command )

inline

Broadcasts a command instance to the listening cluster.

This function must be called on the root whenever the cluster is in listening mode and errs otherwise.

Parameters

command

shared pointer to an instance of a command class derived from mpi_command

Returns

A unique_lock instance locking the mpi_cluster

Definition at line 209 of file mpi_cluster.hpp.

std::vector<int> stan::math::mpi_map_chunks ( std::size_t  num_jobs, std::size_t  chunk_size = 1  )

inline

Maps jobs of given chunk size to workers and returning a vector of counts.

The returned vector is indexed by the rank of each worker and has size equal to the # of workers. Each count per worker is the product of the number of assigned jobs times the chunk size. The jobs are deterministically assigned to workers. This is used for static scheduling of jobs internally.

So with num_workers workers, then the counts for worker with given rank is

chunks[rank] = floor(num_jobs/num_workers) * chunk_size

The remainder jobs num_jobs % num_workers are assigned to rank >=1 workers such that the root (rank = 0) has a little less assigned chunks unless num_jobs < num_workers in which case the first num_jobs nodes receive a job (including the root).

Parameters

num_jobs	Total number of jobs to dispatch
chunk_size	Chunk size per job

Returns

vector indexed by rank with the total number of elements mapped to a given worker

Definition at line 84 of file mpi_cluster.hpp.

template<typename T_y , typename T_covar , typename T_w >

return_type_t<T_y, T_covar, T_w> stan::math::multi_gp_cholesky_lpdf ( const T_y &  y, const T_covar &  L, const T_w &  w  )

inline

Definition at line 87 of file multi_gp_cholesky_lpdf.hpp.

template<typename T_y , typename T_covar , typename T_w >

return_type_t<T_y, T_covar, T_w> stan::math::multi_gp_lpdf ( const T_y &  y, const T_covar &  Sigma, const T_w &  w  )

inline

Definition at line 85 of file multi_gp_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_cholesky_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  L  )

inline

Definition at line 299 of file multi_normal_cholesky_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_covar &	Sigma
	)

Definition at line 19 of file multi_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  Sigma  )

inline

Definition at line 106 of file multi_normal_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_prec_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_covar &	Sigma
	)

Definition at line 20 of file multi_normal_prec_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_prec_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  Sigma  )

inline

Definition at line 106 of file multi_normal_prec_lpdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::multi_student_t_lpdf ( const T_y &  y, const T_dof &  nu, const T_loc &  mu, const T_scale &  Sigma  )

inline

Definition at line 142 of file multi_student_t_lpdf.hpp.

template<typename T_beta , require_eigen_col_vector_t< T_beta > * = nullptr>

return_type_t<T_beta> stan::math::multinomial_logit_lpmf	(	const std::vector< int > &	ns,
		const T_beta &	beta
	)

Definition at line 55 of file multinomial_logit_lpmf.hpp.

template<bool propto, typename T_prob , require_eigen_col_vector_t< T_prob > * = nullptr>

return_type_t<T_prob> stan::math::multinomial_lpmf	(	const std::vector< int > &	ns,
		const T_prob &	theta
	)

Definition at line 17 of file multinomial_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::multinomial_lpmf	(	const std::vector< int > &	ns,
		const T_prob &	theta
	)

Definition at line 45 of file multinomial_lpmf.hpp.

template<class T_theta , class RNG , require_eigen_col_vector_t< T_theta > * = nullptr>

std::vector<int> stan::math::multinomial_rng ( const T_theta &  theta, int  N, RNG &  rng  )

inline

Definition at line 15 of file multinomial_rng.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_vt< is_fvar, Mat1, Mat2 > * = nullptr, require_vt_same< Mat1, Mat2 > * = nullptr, require_not_eigen_row_and_col_t< Mat1, Mat2 > * = nullptr>

auto stan::math::multiply ( const Mat1 &  m1, const Mat2 &  m2  )

inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	first matrix or expression
m2	second matrix or expression

Returns

the product of the first and second matrices

Exceptions

<code>std::invalid_argument</code>

if the number of columns of m1 does not match the number of rows of m2.

Definition at line 18 of file multiply.hpp.

template<typename Mat , typename Scal , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>

auto stan::math::multiply ( const Mat &  m, Scal  c  )

inline

Return specified matrix multiplied by specified scalar.

Template Parameters

Mat	type of the matrix or expression
Scal	type of the scalar

Parameters

m	matrix
c	scalar

Returns

product of matrix and scalar

Definition at line 26 of file multiply.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

auto stan::math::multiply ( T_a &&  A, T_b &&  B  )

inline

Matrix multiplication of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

A	first expression
B	second expression

Returns

Matrix product of given arguments

Definition at line 31 of file multiply.hpp.

template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>

auto stan::math::multiply ( Scal  c, const Mat &  m  )

inline

Return specified scalar multiplied by specified matrix.

Template Parameters

Scal	type of the scalar
Mat	type of the matrix or expression

Parameters

c	scalar
m	matrix

Returns

product of scalar and matrix

Definition at line 43 of file multiply.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_vt< std::is_arithmetic, Mat1, Mat2 > * = nullptr, require_not_eigen_row_and_col_t< Mat1, Mat2 > * = nullptr>

auto stan::math::multiply ( const Mat1 &  m1, const Mat2 &  m2  )

inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	first matrix or expression
m2	second matrix or expression

Returns

the product of the first and second matrices

Exceptions

<code>std::invalid_argument</code>

if the number of columns of m1 does not match the number of rows of m2.

Definition at line 64 of file multiply.hpp.

template<typename T1 , typename T2 , require_stan_scalar_t< T1 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T2 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>

auto stan::math::multiply ( const T1 &  A, T2 &&  B  )

inline

Return matrix multiplied by a scalar.

Template Parameters

T1	type of the scalar
T2	type of the matrix or expression

Parameters

A	scalar
B	matrix

Returns

product of matrix and scalar

Definition at line 79 of file multiply.hpp.

template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_return_type_t< is_var, T1, T2 > * = nullptr, require_row_and_col_vector_t< T1, T2 > * = nullptr>

var stan::math::multiply ( const T1 &  A, const T2 &  B  )

inline

Return the product of a row vector times a column vector as a scalar.

Template Parameters

T1	type of row vector
T2	type of column vector

Parameters

[in]	A	row vector
[in]	B	column vector

Returns

A * B as a scalar

Definition at line 90 of file multiply.hpp.

template<typename T1 , typename T2 , require_stan_scalar_t< T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1 > * = nullptr, require_any_var_t< T1, T2 > * = nullptr>

auto stan::math::multiply ( const T1 &  A, const T2 &  B  )

inline

Return matrix multiplied by a scalar.

Return the product of two matrices.

Template Parameters

T1	type of the matrix or expression
T2	type of the scalar

Parameters

A	matrix
B	scalar

Returns

product of matrix and scalar

This version does not handle row vector times column vector

Template Parameters

T1	type of first matrix
T2	type of second matrix

Parameters

[in]	A	first matrix
[in]	B	second matrix

Returns

A * B

Definition at line 104 of file multiply.hpp.

template<typename RowVec , typename ColVec , require_not_var_t< return_type_t< RowVec, ColVec >> * = nullptr, require_eigen_row_and_col_t< RowVec, ColVec > * = nullptr>

auto stan::math::multiply ( const RowVec &  rv, const ColVec &  v  )

inline

Return the scalar product of the specified row vector and specified column vector.

The return is the same as the dot product. The two vectors must be the same size.

Template Parameters

RowVec	type of the row vector
ColVec	type of the column vector

Parameters

rv	row vector
v	column vector

Returns

scalar result of multiplying row vector by column vector

Exceptions

<code>std::invalid_argument</code>

if rv and v are not the same size

Definition at line 110 of file multiply.hpp.

template<typename T_a , typename T_b , require_stan_scalar_t< T_a > * = nullptr, require_all_kernel_expressions_and_none_scalar_t< T_b > * = nullptr, require_all_not_var_t< T_a, T_b > * = nullptr>

matrix_cl< return_type_t< T_a, T_b > > stan::math::multiply ( const T_a &  a, const T_b &  b  )

inline

Matrix multiplication of a scalar and a kernel generator expressions.

Template Parameters

T_a	type of scalar
T_b	type of the kernel generator expression

Parameters

a	scalar
b	expression

Returns

Matrix product of given arguments

Template Parameters

T_a	type of the kernel generator expression
T_b	type of scalar

Parameters

a	expression
b	scalar

Returns

Matrix product of given arguments

Definition at line 123 of file multiply.hpp.

template<typename Scalar1 , typename Scalar2 , require_all_stan_scalar_t< Scalar1, Scalar2 > * = nullptr>

return_type_t<Scalar1, Scalar2> stan::math::multiply ( Scalar1  m, Scalar2  c  )

inline

Return product of scalars.

Template Parameters

Scalar1	type of first scalar
Scalar2	type of second scalar

Parameters

m	scalar
c	scalar

Returns

product

Definition at line 126 of file multiply.hpp.

template<typename T1 , typename T2 , require_not_matrix_t< T1 > * = nullptr, require_matrix_t< T2 > * = nullptr, require_return_type_t< is_var, T1, T2 > * = nullptr, require_not_row_and_col_vector_t< T1, T2 > * = nullptr>

auto stan::math::multiply ( const T1 &  A, const T2 &  B  )

inline

Return specified matrix multiplied by specified scalar where at least one input has a scalar type of a var_value.

Template Parameters

T1	type of the scalar
T2	type of the matrix or expression

Parameters

A	scalar
B	matrix

Returns

product of matrix and scalar

Template Parameters

T1	type of the matrix or expression
T2	type of the scalar

Parameters

A	matrix
B	scalar

Returns

product of matrix and scalar

Definition at line 140 of file multiply.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 13 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 20 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 26 of file multiply_log.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::multiply_log ( T_a &&  a, T_b &&  b  )

inline

Returns the elementwise multiply_log() of the input.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Elementwise multiply_log() of the input.

Definition at line 28 of file multiply_log.hpp.

template<typename T_a , typename T_b , require_all_arithmetic_t< T_a, T_b > * = nullptr>

return_type_t<T_a, T_b> stan::math::multiply_log ( const T_a  a, const T_b  b  )

inline

Calculate the value of the first argument times log of the second argument while behaving properly with 0 inputs.

\( a * \log b \).

\[ \mbox{multiply\_log}(x, y) = \begin{cases} 0 & \mbox{if } x=y=0\\ x\ln y & \mbox{if } x, y\neq 0 \\[6pt] \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial x} = \begin{cases} \ln y \\[6pt] \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial y} = \begin{cases} \frac{x}{y} \\[6pt] \end{cases} \]

Template Parameters

T_a	type of the first variable
T_b	type of the second variable

Parameters

a	the first variable
b	the second variable

Returns

a * log(b)

Definition at line 49 of file multiply_log.hpp.

var stan::math::multiply_log ( const var &  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to a is log(b). The partial derivative with respect to b is a/b.

Parameters

a	First variable.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 56 of file multiply_log.hpp.

var stan::math::multiply_log ( const var &  a, double  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to a is log(b).

Parameters

a	First variable.
b	Second scalar.

Returns

Value of a*log(b)

Definition at line 69 of file multiply_log.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr, require_all_not_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::multiply_log ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the multiply_log function, when the first and/or second arguments are containers.

Return the product a * log(b).

Return the elementwise product a * log(b).

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

multiply_log function applied to the two inputs.

Both T1 and T2 are matrices, and one of T1 or T2 must be a var_value

Template Parameters

T1	Type of first argument
T2	Type of second argument

Parameters

a	First argument
b	Second argument

Returns

elementwise product of a and log(b)

Template Parameters

T1	Type of matrix argument
T2	Type of scalar argument

Parameters

a	Matrix argument
b	Scalar argument

Returns

Product of a and log(b)

Template Parameters

T1	Type of scalar argument
T2	Type of matrix argument

Parameters

a	Scalar argument
b	Matrix argument

Returns

Product of a and log(b)

Definition at line 70 of file multiply_log.hpp.

var stan::math::multiply_log ( double  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial derivative with respect to b is a/b.

Parameters

a	First scalar.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 82 of file multiply_log.hpp.

template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::RowsAtCompileTime> stan::math::multiply_lower_tri_self_transpose ( const EigMat &  m )

inline

Definition at line 15 of file multiply_lower_tri_self_transpose.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::multiply_lower_tri_self_transpose ( const T &  L )

inline

Definition at line 17 of file multiply_lower_tri_self_transpose.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_autodiff< EigMat > * = nullptr>

matrix_d stan::math::multiply_lower_tri_self_transpose ( const EigMat &  L )

inline

Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose.

Parameters

L	Matrix to multiply.

Returns

The lower triangular values in L times their own transpose.

Definition at line 20 of file multiply_lower_tri_self_transpose.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::multiply_lower_tri_self_transpose ( T_x &&  x )

inline

Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose.

Parameters

x	Matrix to multiply.

Returns

The lower triangular values in x times their own transpose.

Definition at line 21 of file multiply_lower_tri_self_transpose.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::multiply_lower_tri_self_transpose ( const var_value< T > &  A )

inline

Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose.

Parameters

A	Matrix to multiply.

Returns

The lower triangular values in L times their own transpose.

Definition at line 23 of file multiply_lower_tri_self_transpose.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_cdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 24 of file neg_binomial_2_cdf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lccdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 17 of file neg_binomial_2_lccdf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lcdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 18 of file neg_binomial_2_lcdf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >

return_type_t<T_x, T_alpha, T_beta, T_precision> stan::math::neg_binomial_2_log_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta, const T_precision &  phi  )

inline

Definition at line 248 of file neg_binomial_2_log_glm_lpmf.hpp.

template<bool propto, typename T_n , typename T_log_location , typename T_precision , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_log_location, T_precision > * = nullptr>

return_type_t<T_log_location, T_precision> stan::math::neg_binomial_2_log_lpmf	(	const T_n &	n,
		const T_log_location &	eta,
		const T_precision &	phi
	)

Definition at line 28 of file neg_binomial_2_log_lpmf.hpp.

template<typename T_n , typename T_log_location , typename T_precision >

return_type_t<T_log_location, T_precision> stan::math::neg_binomial_2_log_lpmf ( const T_n &  n, const T_log_location &  eta, const T_precision &  phi  )

inline

Definition at line 135 of file neg_binomial_2_log_lpmf.hpp.

template<bool propto, typename T_n , typename T_location , typename T_precision , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_location, T_precision > * = nullptr>

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lpmf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 25 of file neg_binomial_2_lpmf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lpmf ( const T_n &  n, const T_location &  mu, const T_precision &  phi  )

inline

Definition at line 120 of file neg_binomial_2_lpmf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_cdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 25 of file neg_binomial_cdf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lccdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta_param
	)

Definition at line 27 of file neg_binomial_lccdf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lcdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta_param
	)

Definition at line 27 of file neg_binomial_lcdf.hpp.

template<bool propto, typename T_n , typename T_shape , typename T_inv_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_shape, T_inv_scale > * = nullptr>

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lpmf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 33 of file neg_binomial_lpmf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lpmf ( const T_n &  n, const T_shape &  alpha, const T_inv_scale &  beta  )

inline

Definition at line 125 of file neg_binomial_lpmf.hpp.

template<typename T >

std::complex<T> stan::math::neg_i_times ( const std::complex< T > &  z )

inline

Return the specified complex number multiplied by -i.

This compound function is more efficient than mulitplying by the constant -i because it involves only a single arithmetic negation.

Template Parameters

value

type of complex argument

Parameters

[in]

z

complex argument

Returns

argument multipled by -i

Definition at line 36 of file i_times.hpp.

static constexpr double stan::math::negative_infinity ( )

inlinestatic

Return negative infinity.

Returns

Negative infinity.

Definition at line 206 of file constants.hpp.

static size_t stan::math::nested_size ( )

inlinestatic

Definition at line 10 of file nested_size.hpp.

var stan::math::norm ( const std::complex< var > &  z )

inline

Return the squared magnitude of the complex argument.

Parameters

[in]

z

argument

Returns

squared magnitude of the argument

Definition at line 18 of file norm.hpp.

template<typename T >

fvar<T> stan::math::norm ( const std::complex< fvar< T >> &  z )

inline

Return the squared magnitude of the complex argument.

Template Parameters

T	value type of autodiff variable

Parameters

[in]

z

argument

Returns

phase squared magnitude of the argument

Definition at line 19 of file norm.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >

return_type_t<T_y, T_x, T_alpha, T_beta, T_scale> stan::math::normal_id_glm_lpdf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta, const T_scale &  sigma  )

inline

Definition at line 217 of file normal_id_glm_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lccdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 24 of file normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale > * = nullptr>

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lcdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 27 of file normal_lcdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 110 of file normal_lpdf.hpp.

template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >

return_type_t<T_y, T_s, T_loc, T_scale> stan::math::normal_sufficient_lpdf ( const T_y &  y_bar, const T_s &  s_squared, const T_n &  n_obs, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 157 of file normal_sufficient_lpdf.hpp.

static constexpr double stan::math::not_a_number ( )

inlinestatic

Return (quiet) not-a-number.

Returns

Quiet not-a-number.

Definition at line 213 of file constants.hpp.

template<typename T , require_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

size_t stan::math::num_elements

(

const T &

m

)

Returns the number of the elements of a matrix_cl or var_value<matrix_cl<T>>.

Parameters

m	input to determine size of

Returns

number of elements in m

Definition at line 18 of file num_elements.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

int stan::math::num_elements ( const T &  m )

inline

Returns 1, the number of elements in a primitive type.

Returns the size of the specified matrix.

Template Parameters

T	scalar type

Parameters

x	Argument of primitive type.

Returns

1

Template Parameters

T	type of the matrix

Parameters

m	argument matrix

Returns

size of matrix

Definition at line 19 of file num_elements.hpp.

template<typename T >

int stan::math::num_elements ( const std::vector< T > &  v )

inline

Returns the number of elements in the specified vector.

This assumes it is not ragged and that each of its contained elements has the same number of elements.

Template Parameters

T	type of elements in the vector

Parameters

v	argument vector

Returns

number of contained arguments

Definition at line 46 of file num_elements.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_adams	(	const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Adams-Moulton solver in CVODES with defaults for relative_tolerance, absolute_tolerance, and max_num_steps.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 148 of file ode_adams.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_adams_tol	(	const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Adams-Moulton solver from CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance	Relative tolerance passed to CVODES
	absolute_tolerance	Absolute tolerance passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 105 of file ode_adams.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_adams_tol_impl	(	const char *	function_name,
		const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Adams-Moulton solver from CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance	Relative tolerance passed to CVODES
	absolute_tolerance	Absolute tolerance passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 51 of file ode_adams.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr, require_any_not_st_arithmetic< T_y0, T_t0, T_ts, T_Args... > * = nullptr>

auto stan::math::ode_adjoint_impl	(	const char *	function_name,
		F &&	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance_forward,
		const T_abs_tol_fwd &	absolute_tolerance_forward,
		double	relative_tolerance_backward,
		const T_abs_tol_bwd &	absolute_tolerance_backward,
		double	relative_tolerance_quadrature,
		double	absolute_tolerance_quadrature,
		long int	max_num_steps,
		long int	num_steps_between_checkpoints,
		int	interpolation_polynomial,
		int	solver_forward,
		int	solver_backward,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula BDF solver or the non-stiff Adams solver from CVODES. The ODE system is integrated using the adjoint sensitivity approach of CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance_forward	Relative tolerance for forward problem passed to CVODES
	absolute_tolerance_forward	Absolute tolerance per ODE state for forward problem passed to CVODES
	relative_tolerance_backward	Relative tolerance for backward problem passed to CVODES
	absolute_tolerance_backward	Absolute tolerance per ODE state for backward problem passed to CVODES
	relative_tolerance_quadrature	Relative tolerance for quadrature problem passed to CVODES
	absolute_tolerance_quadrature	Absolute tolerance for quadrature problem passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
	num_steps_between_checkpoints	Number of integrator steps after which a checkpoint is stored for the backward pass
	interpolation_polynomial	type of polynomial used for interpolation
	solver_forward	solver used for forward pass
	solver_backward	solver used for backward pass
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

An std::vector of Eigen column vectors with scalars equal to the least upper bound of T_y0, T_t0, T_ts, and the lambda's arguments. This represents the solution to ODE at times ts

Definition at line 71 of file ode_adjoint.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr, require_all_st_arithmetic< T_y0, T_t0, T_ts, T_Args... > * = nullptr>

std::vector<Eigen::VectorXd> stan::math::ode_adjoint_impl	(	const char *	function_name,
		F &&	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance_forward,
		const T_abs_tol_fwd &	absolute_tolerance_forward,
		double	relative_tolerance_backward,
		const T_abs_tol_bwd &	absolute_tolerance_backward,
		double	relative_tolerance_quadrature,
		double	absolute_tolerance_quadrature,
		long int	max_num_steps,
		long int	num_steps_between_checkpoints,
		int	interpolation_polynomial,
		int	solver_forward,
		int	solver_backward,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula BDF solver or the non-stiff Adams solver from CVODES. The ODE system is integrated using the adjoint sensitivity approach of CVODES. This implementation handles the case of a double return type which ensures that no resources are left on the AD stack.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance_forward	Relative tolerance for forward problem passed to CVODES
	absolute_tolerance_forward	Absolute tolerance per ODE state for forward problem passed to CVODES
	relative_tolerance_backward	Relative tolerance for backward problem passed to CVODES
	absolute_tolerance_backward	Absolute tolerance per ODE state for backward problem passed to CVODES
	relative_tolerance_quadrature	Relative tolerance for quadrature problem passed to CVODES
	absolute_tolerance_quadrature	Absolute tolerance for quadrature problem passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
	num_steps_between_checkpoints	Number of integrator steps after which a checkpoint is stored for the backward pass
	interpolation_polynomial	type of polynomial used for interpolation
	solver_forward	solver used for forward pass
	solver_backward	solver used for backward pass
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

An std::vector of Eigen column vectors with scalars equal to the least upper bound of T_y0, T_t0, T_ts, and the lambda's arguments. This represents the solution to ODE at times ts

Definition at line 155 of file ode_adjoint.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename T_abs_tol_fwd , typename T_abs_tol_bwd , typename... T_Args, require_all_eigen_col_vector_t< T_y0, T_abs_tol_fwd, T_abs_tol_bwd > * = nullptr>

auto stan::math::ode_adjoint_tol_ctl	(	F &&	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance_forward,
		const T_abs_tol_fwd &	absolute_tolerance_forward,
		double	relative_tolerance_backward,
		const T_abs_tol_bwd &	absolute_tolerance_backward,
		double	relative_tolerance_quadrature,
		double	absolute_tolerance_quadrature,
		long int	max_num_steps,
		long int	num_steps_between_checkpoints,
		int	interpolation_polynomial,
		int	solver_forward,
		int	solver_backward,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula BDF solver or the non-stiff Adams solver from CVODES. The ODE system is integrated using the adjoint sensitivity approach of CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance_forward	Relative tolerance for forward problem passed to CVODES
	absolute_tolerance_forward	Absolute tolerance per ODE state for forward problem passed to CVODES
	relative_tolerance_backward	Relative tolerance for backward problem passed to CVODES
	absolute_tolerance_backward	Absolute tolerance per ODE state for backward problem passed to CVODES
	relative_tolerance_quadrature	Relative tolerance for quadrature problem passed to CVODES
	absolute_tolerance_quadrature	Absolute tolerance for quadrature problem passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
	num_steps_between_checkpoints	Number of integrator steps after which a checkpoint is stored for the backward pass
	interpolation_polynomial	type of polynomial used for interpolation
	solver_forward	solver used for forward pass
	solver_backward	solver used for backward pass
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

An std::vector of Eigen column vectors with scalars equal to the least upper bound of T_y0, T_t0, T_ts, and the lambda's arguments. This represents the solution to ODE at times ts

Definition at line 243 of file ode_adjoint.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_bdf	(	const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula (BDF) solver in CVODES with defaults for relative_tolerance, absolute_tolerance, and max_num_steps.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 149 of file ode_bdf.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_bdf_tol	(	const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula BDF solver from CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance	Relative tolerance passed to CVODES
	absolute_tolerance	Absolute tolerance passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 106 of file ode_bdf.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... T_Args, require_eigen_col_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, T_Args...>, Eigen::Dynamic, 1> > stan::math::ode_bdf_tol_impl	(	const char *	function_name,
		const F &	f,
		const T_y0 &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const T_Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the stiff backward differentiation formula BDF solver from CVODES.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and not less than t0.
	relative_tolerance	Relative tolerance passed to CVODES
	absolute_tolerance	Absolute tolerance passed to CVODES
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 52 of file ode_bdf.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_ckrk	(	const F &	f,
		const T_y0 &	y0,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using Boost's Cash-Karp Runge-Kutta solver with defaults for relative_tolerance, absolute_tolerance, and max_num_steps.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greather than t0.
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 245 of file ode_ckrk.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_ckrk_tol	(	const F &	f,
		const T_y0 &	y0_arg,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using Boost's Cash-Karp solver.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0_arg	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greater than t0.
	relative_tolerance	Relative tolerance passed to Boost
	absolute_tolerance	Absolute tolerance passed to Boost
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 199 of file ode_ckrk.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_ckrk_tol_impl	(	const char *	function_name,
		const F &	f,
		const T_y0 &	y0_arg,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using Boost's Cash-Karp54 solver.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial condition
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0_arg	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greater than t0.
	relative_tolerance	Relative tolerance passed to Boost
	absolute_tolerance	Absolute tolerance passed to Boost
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 58 of file ode_ckrk.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_rk45	(	const F &	f,
		const T_y0 &	y0,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Runge-Kutta 45 solver in Boost with defaults for relative_tolerance, absolute_tolerance, and max_num_steps.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greather than t0.
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 247 of file ode_rk45.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_rk45_tol	(	const F &	f,
		const T_y0 &	y0_arg,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Runge-Kutta 45 solver in Boost.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	y0_arg	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greater than t0.
	relative_tolerance	Relative tolerance passed to Boost
	absolute_tolerance	Absolute tolerance passed to Boost
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 201 of file ode_rk45.hpp.

template<typename F , typename T_y0 , typename T_t0 , typename T_ts , typename... Args, require_eigen_vector_t< T_y0 > * = nullptr>

std::vector<Eigen::Matrix<stan::return_type_t<T_y0, T_t0, T_ts, Args...>, Eigen::Dynamic, 1> > stan::math::ode_rk45_tol_impl	(	const char *	function_name,
		const F &	f,
		const T_y0 &	y0_arg,
		T_t0	t0,
		const std::vector< T_ts > &	ts,
		double	relative_tolerance,
		double	absolute_tolerance,
		long int	max_num_steps,
		std::ostream *	msgs,
		const Args &...	args
	)

Solve the ODE initial value problem y' = f(t, y), y(t0) = y0 at a set of times, { t1, t2, t3, ...

} using the non-stiff Runge-Kutta 45 solver in Boost.

If the system of equations is stiff, ode_bdf will likely be faster.

f must define an operator() with the signature as: template<typename T_t, typename T_y, typename... T_Args> Eigen::Matrix<stan::return_type_t<T_t, T_y, T_Args...>, Eigen::Dynamic, 1> operator()(const T_t& t, const Eigen::Matrix<T_y, Eigen::Dynamic, 1>& y, std::ostream* msgs, const T_Args&... args);

t is the time, y is the vector-valued state, msgs is a stream for error messages, and args are optional arguments passed to the ODE solve function (which are passed through to f without modification).

Template Parameters

F	Type of ODE right hand side
T_y0	Type of initial condition
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	function_name	Calling function name (for printing debugging messages)
	f	Right hand side of the ODE
	y0_arg	Initial state
	t0	Initial time
	ts	Times at which to solve the ODE at. All values must be sorted and greater than t0.
	relative_tolerance	Relative tolerance passed to Boost
	absolute_tolerance	Absolute tolerance passed to Boost
	max_num_steps	Upper limit on the number of integration steps to take between each output (error if exceeded)
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

Solution to ODE at times ts

Definition at line 59 of file ode_rk45.hpp.

template<typename F , typename T_y0_t0 , typename T_t0 , typename T_t , typename... Args, typename = require_all_arithmetic_t<T_y0_t0, T_t0, T_t, scalar_type_t<Args>...>>

Eigen::VectorXd stan::math::ode_store_sensitivities	(	const F &	f,
		const std::vector< double > &	coupled_state,
		const Eigen::Matrix< T_y0_t0, Eigen::Dynamic, 1 > &	y0,
		T_t0	t0,
		T_t	t,
		std::ostream *	msgs,
		const Args &...	args
	)

When all arguments are arithmetic, there are no sensitivities to store, so the function just returns the current coupled_state.

Template Parameters

F	Type of ODE right hand side
T_y0_t0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	coupled_state	Current state of the coupled_ode_system
	y0	Initial state
	t0	Initial time
	t	Times at which to solve the ODE at
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

ODE state

Definition at line 35 of file ode_store_sensitivities.hpp.

template<typename F , typename T_y0_t0 , typename T_t0 , typename T_t , typename... Args, require_any_autodiff_t< T_y0_t0, T_t0, T_t, scalar_type_t< Args >... > * = nullptr>

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::ode_store_sensitivities	(	const F &	f,
		const std::vector< double > &	coupled_state,
		const Eigen::Matrix< T_y0_t0, Eigen::Dynamic, 1 > &	y0,
		const T_t0 &	t0,
		const T_t &	t,
		std::ostream *	msgs,
		const Args &...	args
	)

Build output vars for a state of the ODE solve, storing the sensitivities precomputed using the forward sensitivity problem in precomputed varis.

Template Parameters

F	Type of ODE right hand side
T_y0_t0	Type of initial state
T_t0	Type of initial time
T_ts	Type of output times
T_Args	Types of pass-through parameters

Parameters

	f	Right hand side of the ODE
	coupled_state	Current state of the coupled_ode_system
	y0	Initial state
	t0	Initial time
	t	Times at which to solve the ODE at
[in,out]	msgs	the print stream for warning messages
	args	Extra arguments passed unmodified through to ODE right hand side

Returns

ODE state with scalar type var

Definition at line 36 of file ode_store_sensitivities.hpp.

template<typename T , typename M , typename S , require_all_kernel_expressions_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of unconstrained input
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 38 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr, require_any_var_t< T, M, S > * = nullptr>

var_value<matrix_cl<double> > stan::math::offset_multiplier_constrain ( T &&  A, M &&  mu, S &&  sigma  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of unconstrained input
M	type of offset
S	type of multiplier

Parameters

[in]	A	Unconstrained input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 40 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of scalar
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained scalar input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 43 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_kernel_expressions_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma, double &  lp  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of unconstrained input
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output
[in,out]	lp	Reference to log probability to increment.

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 79 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr, require_any_not_stan_scalar_t< T, M, S > * = nullptr, require_any_var_t< T, M, S > * = nullptr>

var_value<matrix_cl<double> > stan::math::offset_multiplier_constrain ( T &&  A, M &&  mu, S &&  sigma, var &  lp  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of unconstrained input
M	type of offset
S	type of multiplier

Parameters

[in]	A	Unconstrained input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output
[in,out]	lp	Reference to log probability to increment.

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 87 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T, M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma, return_type_t< T, M, S > &  lp  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier, incrementing the specified reference with the log absolute Jacobian determinant of the transform.

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and multiplier is one, this function reduces to identity_constraint(x, lp).

Template Parameters

T	type of scalar
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained scalar input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output
[in,out]	lp	Reference to log probability to increment.

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
M	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
S	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

[in]	x	Unconstrained scalar input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output
[in,out]	lp	log density accumulator

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 92 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const M &  mu, const S &  sigma  )

inline

Overload for array of x and non-array mu and sigma.

Definition at line 122 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const M &  mu, const S &  sigma, return_type_t< T, M, S > &  lp  )

inline

Overload for array of x and non-array mu and sigma with lp.

Definition at line 141 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const M &  mu, const std::vector< S > &  sigma  )

inline

Overload for array of x and sigma and non-array mu.

Definition at line 161 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const M &  mu, const std::vector< S > &  sigma, return_type_t< T, M, S > &  lp  )

inline

Overload for array of x and sigma and non-array mu with lp.

Definition at line 180 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const std::vector< M > &  mu, const S &  sigma  )

inline

Overload for array of x and mu and non-array sigma.

Definition at line 200 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const std::vector< M > &  mu, const S &  sigma, return_type_t< T, M, S > &  lp  )

inline

Overload for array of x and mu and non-array sigma with lp.

Definition at line 220 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S >

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const std::vector< M > &  mu, const std::vector< S > &  sigma  )

inline

Overload for array of x, mu, and sigma.

Definition at line 240 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S >

auto stan::math::offset_multiplier_constrain ( const std::vector< T > &  x, const std::vector< M > &  mu, const std::vector< S > &  sigma, return_type_t< T, M, S > &  lp  )

inline

Overload for array of x, mu, and sigma with lp.

Definition at line 259 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S >

auto stan::math::offset_multiplier_free ( const T &  y, const M &  mu, const S &  sigma  )

inline

Return the unconstrained variable that transforms to the specified offset and multiplier constrained variable given the specified offset and multiplier.

The transform in locmultiplier_constrain(T, double, double), is reversed by the reverse affine transformation,

\(f^{-1}(y) = \frac{y - L}{S}\)

where \(L\) and \(S\) are the offset and multiplier.

If the offset is zero and multiplier is one, this function reduces to identity_free(y).

Template Parameters

T	type of constrained variable
L	type of offset
S	type of multiplier

Parameters

	y	constrained value
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

the unconstrained variable that transforms to the given constrained variable given the offset and multiplier

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite
std::invalid_argument	if non-scalar arguments don't match in size

Definition at line 42 of file offset_multiplier_free.hpp.

template<typename T , typename M , typename S , require_all_not_std_vector_t< M, S > * = nullptr>

auto stan::math::offset_multiplier_free ( const std::vector< T > &  x, const M &  mu, const S &  sigma  )

inline

Overload for array of x and non-array mu and sigma.

Definition at line 66 of file offset_multiplier_free.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< M > * = nullptr>

auto stan::math::offset_multiplier_free ( const std::vector< T > &  x, const M &  mu, const std::vector< S > &  sigma  )

inline

Overload for array of x and sigma and non-array mu.

Definition at line 84 of file offset_multiplier_free.hpp.

template<typename T , typename M , typename S , require_not_std_vector_t< S > * = nullptr>

auto stan::math::offset_multiplier_free ( const std::vector< T > &  x, const std::vector< M > &  mu, const S &  sigma  )

inline

Overload for array of x and mu and non-array sigma.

Definition at line 103 of file offset_multiplier_free.hpp.

template<typename T , typename M , typename S >

auto stan::math::offset_multiplier_free ( const std::vector< T > &  x, const std::vector< M > &  mu, const std::vector< S > &  sigma  )

inline

Overload for array of x, mu, and sigma.

Definition at line 121 of file offset_multiplier_free.hpp.

std::vector<double> stan::math::one_hot_array ( int  K, int  k  )

inline

Return an array with 1 in the k-th position and zero elsewhere.

Parameters

K	size of the array
k	position of the 1 (indexing from 1)

Returns

An array of size K with all elements initialised to zero and a 1 in the k-th position.

Exceptions

std::domain_error

if K is not positive, or if k is less than 1 or greater than K.

Definition at line 20 of file one_hot_array.hpp.

std::vector<int> stan::math::one_hot_int_array ( int  K, int  k  )

inline

Return an integer array with 1 in the k-th position and zero elsewhere.

Parameters

K	size of the array
k	position of the 1 (indexing from 1)

Returns

An integer array of size K with all elements initialised to zero and a 1 in the k-th position.

Exceptions

std::domain_error

if K is not positive, or if k is less than 1 or greater than K.

Definition at line 20 of file one_hot_int_array.hpp.

Eigen::RowVectorXd stan::math::one_hot_row_vector ( int  K, int  k  )

inline

Return a row vector with 1 in the k-th position and zero elsewhere.

Parameters

K	size of the row vector
k	position of the 1 (indexing from 1)

Returns

A row vector of size K with all elements initialised to zero and a 1 in the k-th position.

Exceptions

std::domain_error

if K is not positive, or if k is less than 1 or greater than K.

Definition at line 20 of file one_hot_row_vector.hpp.

Eigen::VectorXd stan::math::one_hot_vector ( int  K, int  k  )

inline

Return a vector with 1 in the k-th position and zero elsewhere.

Parameters

K	size of the vector
k	position of the 1 (indexing from 1)

Returns

A vector of size K with all elements initialised to zero and a 1 in the k-th position.

Exceptions

std::domain_error

if K is not positive, or if k is less than 1 or greater than K.

Definition at line 20 of file one_hot_vector.hpp.

std::vector<double> stan::math::ones_array ( int  K )

inline

Return an array of ones.

Parameters

K	size of the array

Returns

An array of size K with all elements initialised to 1.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file ones_array.hpp.

std::vector<int> stan::math::ones_int_array ( int  K )

inline

Return an integer array of ones.

Parameters

K	size of the array

Returns

An integer array of size K with all elements initialised to 1.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file ones_int_array.hpp.

auto stan::math::ones_row_vector ( int  K )

inline

Return a row vector of ones.

Parameters

K	size of the row vector

Returns

A row vector of size K with all elements initialised to 1.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file ones_row_vector.hpp.

auto stan::math::ones_vector ( int  K )

inline

Return a vector of ones.

Parameters

K	size of the vector

Returns

A vector of size K with all elements initialised to 1.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file ones_vector.hpp.

bool stan::math::operator! ( const var &  x )

inline

Return the negation of the value of the argument as defined by !.

Parameters

[in]

x

argument

Returns

negation of argument value

Definition at line 17 of file operator_unary_not.hpp.

template<typename T >

bool stan::math::operator! ( const fvar< T > &  x )

inline

Return the negation of the value of the argument as defined by !.

Template Parameters

value

and tangent type for variables

Parameters

[in]

x

argument

Returns

negation of argument value

Definition at line 18 of file operator_unary_not.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

logical_negation_<as_operation_cl_t<T> > stan::math::operator! ( T &&  a )

inline

Logical negation of a kernel generator expression.

Template Parameters

T	type of the argument

Parameters

a	argument expression

Returns

logical negation of given expression

Definition at line 109 of file unary_operation_cl.hpp.

template<typename T >

bool stan::math::operator!= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the value of the two arguments are not equal as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the values of the arguments are not equal

Definition at line 19 of file operator_not_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator!= ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return true if the complex numbers have unequal imaginary or complex parts.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are equal

Definition at line 21 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var &  a, const var &  b  )

inline

Inequality operator comparing two variables' values (C++).

\[ \mbox{operator!=}(x, y) = \begin{cases} 0 & \mbox{if } x = y\\ 1 & \mbox{if } x \neq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if the first variable's value is not the same as the second's.

Definition at line 30 of file operator_not_equal.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator!=	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 32 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator!= ( const fvar< T > &  x, double  y  )

inline

Return true if the value of the first argument is not equal to the second argument as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the value of the first argument is not equal to the second argument

Definition at line 34 of file operator_not_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator!= ( const std::complex< U > &  x, const V &  y  )

inline

Return true if the first argument's real part is unequal to the second argument or the first argument's imaginary part is unequal to zero.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are equal

Definition at line 37 of file operator_not_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator!= ( const var &  a, Arith  b  )

inline

Inequality operator comparing a variable's value and a double (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if the first variable's value is not the same as the second value.

Definition at line 45 of file operator_not_equal.hpp.

template<typename T >

bool stan::math::operator!= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is not equal to the value of the second argument as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is not equal to the value of the second argument

Definition at line 49 of file operator_not_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator!= ( const U &  x, const std::complex< V > &  y  )

inline

Return true if the first argument is unequal to the real part of the second argument or the imaginary part of the second argument is nonzero.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are not equal

Definition at line 53 of file operator_not_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator!= ( Arith  a, const var &  b  )

inline

Inequality operator comparing a double and a variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First value.
b	Second variable.

Returns

True if the first value is not the same as the second variable's value.

Definition at line 60 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var &  x, const std::complex< var > &  z  )

inline

Return false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero.

Parameters

x	real number
z	complex number

Returns

false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero

Definition at line 73 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const std::complex< var > &  z, const var &  y  )

inline

Return false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero.

Parameters

z	complex number
y	real number

Returns

false if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero

Definition at line 86 of file operator_not_equal.hpp.

bool stan::math::operator&& ( const var &  x, const var &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of the arguments' values

Definition at line 18 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of the arguments' values

Definition at line 19 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( const fvar< T > &  x, double  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument's value and second argument

Definition at line 34 of file operator_logical_and.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator&& ( const var &  x, Arith  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

Var	value type of a var
Arith	An arithmetic type

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument's value and second argument

Definition at line 34 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( double  x, const fvar< T > &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument and the second argument's value

Definition at line 49 of file operator_logical_and.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator&& ( Arith  x, const var &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

Var	value type of a var
Arith	An arithmetic type

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument and second argument's value

Definition at line 50 of file operator_logical_and.hpp.

template<typename T >

fvar<T> stan::math::operator* ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 18 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::math::operator* ( double  x, const fvar< T > &  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 31 of file operator_multiplication.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator* ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return the product of the arguments.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

product of the arguments

Definition at line 39 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::math::operator* ( const fvar< T > &  x, double  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 44 of file operator_multiplication.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator* ( const std::complex< U > &  x, const V &  y  )

inline

Return the product of the arguments.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

product of the arguments

Definition at line 54 of file operator_multiplication.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

auto stan::math::operator* ( const T_a &  A, const T_b &  B  )

inline

Matrix multiplication of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

A	first expression
B	second expression

Returns

Matrix product of given arguments

Definition at line 62 of file multiply.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator* ( const U &  x, const std::complex< V > &  y  )

inline

Return the product of the arguments.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

product of the arguments

Definition at line 68 of file operator_multiplication.hpp.

var stan::math::operator* ( const var &  a, const var &  b  )

inline

Multiplication operator for two variables (C++).

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) = y\), and

\(\frac{\partial}{\partial y} (x * y) = x\).

\[ \mbox{operator*}(x, y) = \begin{cases} xy & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of multiplying operands.

Definition at line 71 of file operator_multiplication.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator* ( const var &  a, Arith  b  )

inline

Multiplication operator for a variable and a scalar (C++).

The partial derivative for the variable is

\(\frac{\partial}{\partial x} (x * c) = c\), and

Template Parameters

Arith

An arithmetic type

Parameters

a	Variable operand.
b	Scalar operand.

Returns

Variable result of multiplying operands.

Definition at line 88 of file operator_multiplication.hpp.

template<typename T_a , typename T_b , typename = require_all_kernel_expressions_and_none_scalar_t<T_a, T_b>>

matrix_cl<return_type_t<T_a, T_b> > stan::math::operator* ( const T_a &  a, const T_b &  b  )

inline

Matrix multiplication of two kernel generator expressions.

Evaluates both expressions before calculating the matrix product.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Matrix product of given arguments

Definition at line 106 of file multiply.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator* ( Arith  a, const var &  b  )

inline

Multiplication operator for a scalar and a variable (C++).

The partial derivative for the variable is

\(\frac{\partial}{\partial y} (c * y) = c\).

Template Parameters

Arith

An arithmetic type

Parameters

a	Scalar operand.
b	Variable operand.

Returns

Variable result of multiplying the operands.

Definition at line 108 of file operator_multiplication.hpp.

template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>

T1 stan::math::operator*=	(	T1 &&	a,
		T2 &&	b
	)

Definition at line 24 of file compound_assignments.hpp.

template<typename U , require_autodiff_t< U > >

std::complex<U> stan::math::operator+ ( const std::complex< U > &  x )

inline

Return the argument.

Template Parameters

U	value type argument

Parameters

x

argument

Returns

argument

Definition at line 18 of file operator_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the sum of the specified forward mode addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 18 of file operator_addition.hpp.

template<typename T , require_var_vt< is_kernel_expression_and_not_scalar, T > * = nullptr>

T stan::math::operator+ ( T &&  M )

inline

Returns the unary plus of the input.

Parameters

M	input kernel expression

Returns

result of unary plus of the input.

Definition at line 19 of file operator_unary_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x )

inline

Returns the argument.

It is included for completeness. The primitive unary operator+ exists to promote integer to floating point values.

Template Parameters

T	value and tangent type of the argument

Parameters

x

argument

Returns

the argument

Definition at line 19 of file operator_unary_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( double  x1, const fvar< T > &  x2  )

inline

Return the sum of the specified double and forward mode addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 31 of file operator_addition.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator+ ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return the sum of the arguments.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

sum of the arguments

Definition at line 39 of file operator_addition.hpp.

var stan::math::operator+ ( const var &  a )

inline

Unary plus operator for variables (C++).

The function simply returns its input, because

\(\frac{d}{dx} +x = \frac{d}{dx} x = 1\).

The effect of unary plus on a built-in C++ scalar type is integer promotion. Because variables are all double-precision floating point already, promotion is not necessary.

\[ \mbox{operator+}(x) = \begin{cases} x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Argument variable.

Returns

The input reference.

Definition at line 44 of file operator_unary_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x1, double  x2  )

inline

Return the sum of the specified forward mode and double addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 44 of file operator_addition.hpp.

var stan::math::operator+ ( const var &  a, const var &  b  )

inline

Addition operator for variables (C++).

The partial derivatives are defined by

\(\frac{\partial}{\partial x} (x+y) = 1\), and

\(\frac{\partial}{\partial y} (x+y) = 1\).

\[ \mbox{operator+}(x, y) = \begin{cases} x+y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial y} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 53 of file operator_addition.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator+ ( const std::complex< U > &  x, const V &  y  )

inline

Return the sum of the arguments.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

sum of the arguments

Definition at line 54 of file operator_addition.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

auto stan::math::operator+ ( const T_a &  a, const T_b &  b  )

inline

Addition of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

The sum of the given arguments

Definition at line 55 of file add.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator+ ( const U &  x, const std::complex< V > &  y  )

inline

Return the sum of the arguments.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

sum of the arguments

Definition at line 68 of file operator_addition.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator+ ( const var &  a, Arith  b  )

inline

Addition operator for variable and scalar (C++).

The derivative with respect to the variable is

\(\frac{d}{dx} (x + c) = 1\).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable operand.
b	Second scalar operand.

Returns

Result of adding variable and scalar.

Definition at line 74 of file operator_addition.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator+ ( Arith  a, const var &  b  )

inline

Addition operator for scalar and variable (C++).

The derivative with respect to the variable is

\(\frac{d}{dy} (c + y) = 1\).

Template Parameters

Arith

An arithmetic type

Parameters

a	First scalar operand.
b	Second variable operand.

Returns

Result of adding variable and scalar.

Definition at line 96 of file operator_addition.hpp.

template<typename VarMat1 , typename VarMat2 , require_any_var_matrix_t< VarMat1, VarMat2 > * = nullptr>

auto stan::math::operator+ ( const VarMat1 &  a, const VarMat2 &  b  )

inline

Addition operator for matrix variables.

Template Parameters

VarMat1	A matrix of vars or a var with an underlying matrix type.
VarMat2	A matrix of vars or a var with an underlying matrix type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 277 of file operator_addition.hpp.

var& stan::math::operator++ ( var &  a )

inline

Prefix increment operator for variables (C++).

Following C++, (++a) is defined to behave exactly as (a = a + 1.0) does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to increment.

Returns

Reference the result of incrementing this input variable.

Definition at line 22 of file operator_unary_increment.hpp.

var stan::math::operator++ ( var &  a, int    )

inline

Postfix increment operator for variables (C++).

Following C++, the expression (a++) is defined to behave like the sequence of operations

var temp = a; a = a + 1.0; return temp;

Parameters

a	Variable to increment.

Returns

Input variable.

Definition at line 38 of file operator_unary_increment.hpp.

template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>

T1 stan::math::operator+=	(	T1 &&	a,
		T2 &&	b
	)

Definition at line 22 of file compound_assignments.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x )

inline

Return the negation of the specified argument.

Template Parameters

value

and tangent type of the argument

Parameters

[in]

x

argument

Returns

negation of argument

Definition at line 17 of file operator_unary_minus.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 18 of file operator_subtraction.hpp.

template<typename U , require_autodiff_t< U > >

std::complex<U> stan::math::operator- ( const std::complex< U > &  x )

inline

Return the negation of the argument.

Template Parameters

U	value type argument

Parameters

x

argument

Returns

negation of the argument

Definition at line 18 of file operator_minus.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::operator- ( const var_value< T > &  M )

inline

Returns the unary minus of the input.

Parameters

M	input kernel expression

Returns

result of unary minus of the input.

Definition at line 19 of file operator_unary_minus.hpp.

template<typename T >

fvar<T> stan::math::operator- ( double  x1, const fvar< T > &  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 31 of file operator_subtraction.hpp.

var stan::math::operator- ( const var &  a )

inline

Unary negation operator for variables (C++).

\(\frac{d}{dx} -x = -1\).

\[ \mbox{operator-}(x) = \begin{cases} -x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Argument variable.

Returns

Negation of variable.

Definition at line 37 of file operator_unary_negative.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator- ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return the difference of the arguments.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

difference of the arguments

Definition at line 39 of file operator_subtraction.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x1, double  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 44 of file operator_subtraction.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::operator- ( const T &  a )

inline

Compute additive inverse of input.

Template Parameters

T	type of input

Parameters

a

input

Returns

negative of input

Definition at line 50 of file operator_unary_negative.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator- ( const std::complex< U > &  x, const V &  y  )

inline

Return the difference of the arguments.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

difference of the arguments

Definition at line 54 of file operator_subtraction.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

auto stan::math::operator- ( const T_a &  a, const T_b &  b  )

inline

Subtraction of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

The subtraction of the the second argument from the first

Definition at line 55 of file subtract.hpp.

var stan::math::operator- ( const var &  a, const var &  b  )

inline

Subtraction operator for variables.

The partial derivatives are defined by

\(\frac{\partial}{\partial x} (x-y) = 1\), and

\(\frac{\partial}{\partial y} (x-y) = -1\).

\[ \mbox{operator-}(x, y) = \begin{cases} x-y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

Var1	value type of a var
Var2	value type of a var

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting the second variable from the first.

Definition at line 56 of file operator_subtraction.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator- ( const U &  x, const std::complex< V > &  y  )

inline

Return the difference of the arguments.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

difference of the arguments

Definition at line 68 of file operator_subtraction.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator- ( const var &  a, Arith  b  )

inline

Subtraction operator for variable and scalar.

The derivative for the variable is

\(\frac{\partial}{\partial x} (x-c) = 1\), and

Template Parameters

Var	value type of a var
Arith	An arithmetic type

Parameters

a	First variable operand.
b	Second scalar operand.

Returns

Result of subtracting the scalar from the variable.

Definition at line 78 of file operator_subtraction.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator- ( Arith  a, const var &  b  )

inline

Subtraction operator for scalar and variable.

The derivative for the variable is

\(\frac{\partial}{\partial y} (c-y) = -1\), and

Template Parameters

Var	value type of a var
Arith	An arithmetic type

Parameters

a	First scalar operand.
b	Second variable operand.

Returns

Result of subtracting a variable from a scalar.

Definition at line 101 of file operator_subtraction.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

unary_minus_<as_operation_cl_t<T> > stan::math::operator- ( T &&  a )

inline

Unary minus of a kernel generator expression.

Template Parameters

T	type of the argument

Parameters

a	argument expression

Returns

unary minus of given expression

Definition at line 162 of file unary_operation_cl.hpp.

template<typename VarMat1 , typename VarMat2 , require_any_var_matrix_t< VarMat1, VarMat2 > * = nullptr>

auto stan::math::operator- ( const VarMat1 &  a, const VarMat2 &  b  )

inline

Addition operator for matrix variables.

Template Parameters

VarMat1	A matrix of vars or a var with an underlying matrix type.
VarMat2	A matrix of vars or a var with an underlying matrix type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 309 of file operator_subtraction.hpp.

var& stan::math::operator-- ( var &  a )

inline

Prefix decrement operator for variables (C++).

Following C++, (–a) is defined to behave exactly as

a = a - 1.0)

does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to decrement.

Returns

Reference the result of decrementing this input variable.

Definition at line 26 of file operator_unary_decrement.hpp.

var stan::math::operator-- ( var &  a, int    )

inline

Postfix decrement operator for variables (C++).

Following C++, the expression (a–) is defined to behave like the sequence of operations

var temp = a; a = a - 1.0; return temp;

Parameters

a	Variable to decrement.

Returns

Input variable.

Definition at line 42 of file operator_unary_decrement.hpp.

template<typename T1 , typename T2 , require_kernel_expression_lhs_t< T1 > * = nullptr, require_all_kernel_expressions_t< T2 > * = nullptr>

T1 stan::math::operator-=	(	T1 &&	a,
		T2 &&	b
	)

Definition at line 23 of file compound_assignments.hpp.

template<typename T >

fvar<T> stan::math::operator/ ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 21 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

fvar<T> stan::math::operator/ ( const fvar< T > &  x1, U  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 35 of file operator_division.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator/ ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return the quotient of the arguments.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

quotient of the arguments

Definition at line 39 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

fvar<T> stan::math::operator/ ( U  x1, const fvar< T > &  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 49 of file operator_division.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator/ ( const std::complex< U > &  x, const V &  y  )

inline

Return the quotient of the arguments.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

quotient of the arguments

Definition at line 54 of file operator_division.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< fvar< T >> &  x1, const std::complex< fvar< T >> &  x2  )

inline

Definition at line 55 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< fvar< T >> &  x1, const std::complex< U > &  x2  )

inline

Definition at line 60 of file operator_division.hpp.

var stan::math::operator/ ( const var &  dividend, const var &  divisor  )

inline

Division operator for two variables (C++).

The partial derivatives for the variables are

\(\frac{\partial}{\partial x} (x/y) = 1/y\), and

\(\frac{\partial}{\partial y} (x/y) = -x / y^2\).

\[ \mbox{operator/}(x, y) = \begin{cases} \frac{x}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial x} = \begin{cases} \frac{1}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial y} = \begin{cases} -\frac{x}{y^2} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

dividend	First variable operand.
divisor	Second variable operand.

Returns

Variable result of dividing the first variable by the second.

Definition at line 61 of file operator_division.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< fvar< T >> &  x1, const fvar< T > &  x2  )

inline

Definition at line 65 of file operator_division.hpp.

template<typename U , typename V >

complex_return_t<U, V> stan::math::operator/ ( const U &  x, const std::complex< V > &  y  )

inline

Return the quotient of the arguments.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

quotient of the arguments

Definition at line 68 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< fvar< T >> &  x1, U  x2  )

inline

Definition at line 70 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< U > &  x1, const std::complex< fvar< T >> &  x2  )

inline

Definition at line 75 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

std::complex<fvar<T> > stan::math::operator/ ( const std::complex< U > &  x1, const fvar< T > &  x2  )

inline

Definition at line 80 of file operator_division.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator/ ( const var &  dividend, Arith  divisor  )

inline

Division operator for dividing a variable by a scalar (C++).

The derivative with respect to the variable is

\(\frac{\partial}{\partial x} (x/c) = 1/c\).

Template Parameters

Var	value type of a var
Arith	An arithmetic type

Parameters

dividend	Variable operand.
divisor	Scalar operand.

Returns

Variable result of dividing the variable by the scalar.

Definition at line 84 of file operator_division.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::operator/ ( const fvar< T > &  x1, const std::complex< fvar< T >> &  x2  )

inline

Definition at line 86 of file operator_division.hpp.

template<typename T , typename U , typename = std::enable_if_t<std::is_arithmetic<U>::value>>

std::complex<fvar<T> > stan::math::operator/ ( const fvar< T > &  x1, const std::complex< U > &  x2  )

inline

Definition at line 92 of file operator_division.hpp.

template<typename T , typename U , require_arithmetic_t< U > * = nullptr>

std::complex<fvar<T> > stan::math::operator/ ( U  x1, const std::complex< fvar< T >> &  x2  )

inline

Definition at line 98 of file operator_division.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

var stan::math::operator/ ( Arith  dividend, const var &  divisor  )

inline

Division operator for dividing a scalar by a variable (C++).

The derivative with respect to the variable is

\(\frac{d}{d y} (c/y) = -c / y^2\).

Template Parameters

Arith

An arithmetic type

Parameters

dividend	Scalar operand.
divisor	Variable operand.

Returns

Quotient of the dividend and divisor.

Definition at line 106 of file operator_division.hpp.

template<typename T1 , typename T2 , require_any_var_matrix_t< T1, T2 > * = nullptr>

auto stan::math::operator/ ( const T1 &  dividend, const T2 &  divisor  )

inline

Definition at line 255 of file operator_division.hpp.

std::complex<var> stan::math::operator/ ( const std::complex< var > &  x1, const std::complex< var > &  x2  )

inline

Definition at line 259 of file operator_division.hpp.

template<typename T >

bool stan::math::operator< ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a value less than the value of the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than the second argument's value

Definition at line 20 of file operator_less_than.hpp.

bool stan::math::operator< ( const var &  a, const var &  b  )

inline

Less than operator comparing variables' values (C++).

\[ \mbox{operator\textless}(x, y) = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is less than second's.

Definition at line 25 of file operator_less_than.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator<	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 27 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator< ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is less than the value of the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is less than the second's value argument

Definition at line 35 of file operator_less_than.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator< ( const var &  a, Arith  b  )

inline

Less than operator comparing variable's value and a double (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is less than second value.

Definition at line 37 of file operator_less_than.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator< ( Arith  a, const var &  b  )

inline

Less than operator comparing a double and variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First value.
b	Second variable.

Returns

True if first value is less than second variable's value.

Definition at line 51 of file operator_less_than.hpp.

template<typename T >

bool stan::math::operator< ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a value less than the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value less than the second argument argument

Definition at line 51 of file operator_less_than.hpp.

template<typename T >

bool stan::math::operator<= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a value less than or equal to the value of the second argument as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than or equal to the second argument's value

Definition at line 21 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const var &  a, const var &  b  )

inline

Less than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textless=}(x, y) = \begin{cases} 0 & \mbox{if } x > y\\ 1 & \mbox{if } x \leq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is less than or equal to the second's.

Definition at line 27 of file operator_less_than_or_equal.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator<=	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 28 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator<= ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a value less than or equal to the second argument as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than or equal to the second argument

Definition at line 36 of file operator_less_than_or_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator<= ( const var &  a, Arith  b  )

inline

Less than or equal operator comparing a variable's value and a scalar (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is less than or equal to the second value.

Definition at line 42 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::math::operator<= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is less than or equal to the second argument's value as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is less than or equal to the second argument's value

Definition at line 51 of file operator_less_than_or_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator<= ( Arith  a, const var &  b  )

inline

Less than or equal operator comparing a double and variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First value.
b	Second variable.

Returns

True if first value is less than or equal to the second variable's value.

Definition at line 57 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::math::operator== ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the specified variables have equal values as defined by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 19 of file operator_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator== ( const std::complex< U > &  x, const std::complex< V > &  y  )

inline

Return true if the complex numbers have equal imaginary and complex parts.

Template Parameters

U	value type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are equal

Definition at line 21 of file operator_equal_equal.hpp.

bool stan::math::operator== ( const var &  a, const var &  b  )

inline

Equality operator comparing two variables' values (C++).

\[ \mbox{operator==}(x, y) = \begin{cases} 0 & \mbox{if } x \neq y\\ 1 & \mbox{if } x = y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if the first variable's value is the same as the second's.

Definition at line 27 of file operator_equal.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator==	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 31 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator== ( const fvar< T > &  x, double  y  )

inline

Return true if the the first variable has a value equal to the second argument as defined by by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 34 of file operator_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator== ( const std::complex< U > &  x, const V &  y  )

inline

Return true if the first argument's real part is equal to the second argument and the first argument's imaginary part is zero.

Template Parameters

U	value type of first argument
V	type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are equal

Definition at line 36 of file operator_equal_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator== ( const var &  a, Arith  b  )

inline

Equality operator comparing a variable's value and a double (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if the first variable's value is the same as the second value.

Definition at line 42 of file operator_equal.hpp.

template<typename T >

bool stan::math::operator== ( double  x, const fvar< T > &  y  )

inline

Return true if the the first argument is equal to the value of the second argument as defined by by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 48 of file operator_equal.hpp.

template<typename U , typename V , typename = require_any_autodiff_t<U, V>>

bool stan::math::operator== ( const U &  x, const std::complex< V > &  y  )

inline

Return true if the first argument is equal to the real part of the second argument and the imaginary part of the second argument is zero.

Template Parameters

U	type of first argument
V	value type of second argument

Parameters

x	first argument
y	second argument

Returns

true if the arguments are equal

Definition at line 52 of file operator_equal_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator== ( Arith  a, const var &  b  )

inline

Equality operator comparing a scalar and a variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First scalar.
b	Second variable.

Returns

True if the variable's value is equal to the scalar.

Definition at line 56 of file operator_equal.hpp.

bool stan::math::operator== ( const var &  x, const std::complex< var > &  z  )

inline

Return true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero.

Parameters

x	real number
z	complex number

Returns

true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero

Definition at line 69 of file operator_equal.hpp.

bool stan::math::operator== ( const std::complex< var > &  z, const var &  y  )

inline

Return true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero.

Parameters

z	complex number
y	real number

Returns

true if the real number is equal to the real part of the complex number, and the imaginary part of the complex number is zero

Definition at line 82 of file operator_equal.hpp.

template<typename T >

bool stan::math::operator> ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 20 of file operator_greater_than.hpp.

bool stan::math::operator> ( const var &  a, const var &  b  )

inline

Greater than operator comparing variables' values (C++).

\[ \mbox{operator\textgreater}(x, y) = \begin{cases} 0 & \mbox{if } x \leq y\\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is greater than second's.

Definition at line 26 of file operator_greater_than.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator>	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 29 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator> ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 35 of file operator_greater_than.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator> ( const var &  a, Arith  b  )

inline

Greater than operator comparing variable's value and double (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is greater than second value.

Definition at line 38 of file operator_greater_than.hpp.

template<typename T >

bool stan::math::operator> ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 50 of file operator_greater_than.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator> ( Arith  a, const var &  b  )

inline

Greater than operator comparing a double and a variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First value.
b	Second variable.

Returns

True if first value is greater than second variable's value.

Definition at line 52 of file operator_greater_than.hpp.

template<typename T >

bool stan::math::operator>= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the value of the first argument is greater than or equal to that of the second as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 20 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const var &  a, const var &  b  )

inline

Greater than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textgreater=}(x, y) = \begin{cases} 0 & \mbox{if } x < y\\ 1 & \mbox{if } x \geq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is greater than or equal to the second's.

Definition at line 28 of file operator_greater_than_or_equal.hpp.

template<typename T_a , typename T_b , require_any_eigen_t< T_a, T_b > * = nullptr, require_any_st_autodiff< T_a, T_b > * = nullptr, require_not_st_same< T_a, T_b > * = nullptr>

auto stan::math::operator>=	(	const T_a &	a,
		const T_b &	b
	)

Definition at line 30 of file eigen_comparisons.hpp.

template<typename T >

bool stan::math::operator>= ( const fvar< T > &  x, double  y  )

inline

Return true if the value of the first argument has a value greater than or equal to the second argument as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 36 of file operator_greater_than_or_equal.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator>= ( const var &  a, Arith  b  )

inline

Greater than or equal operator comparing variable's value and double (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is greater than or equal to second value.

Definition at line 43 of file operator_greater_than_or_equal.hpp.

template<typename T >

bool stan::math::operator>= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is greater than or equal to the value of the second argument as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 52 of file operator_greater_than_or_equal.hpp.

template<typename Arith , typename Var , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator>= ( Arith  a, const var &  b  )

inline

Greater than or equal operator comparing double and variable's value (C++).

Template Parameters

Arith

An arithmetic type

Parameters

a	First value.
b	Second variable.

Returns

True if the first value is greater than or equal to the second variable's value.

Definition at line 58 of file operator_greater_than_or_equal.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::ColsAtCompileTime> stan::math::operator^ ( const EigMat &  M, const int  n  )

inline

Definition at line 51 of file matrix_power.hpp.

bool stan::math::operator|| ( const var &  x, const var &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of the arguments' values

Definition at line 18 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of the arguments' values

Definition at line 19 of file operator_logical_or.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator|| ( const var &  x, Arith  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

Arith

An arithmetic type

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument's value and second argument

Definition at line 33 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( const fvar< T > &  x, double  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument's value and second argument

Definition at line 34 of file operator_logical_or.hpp.

template<typename Arith , require_arithmetic_t< Arith > * = nullptr>

bool stan::math::operator|| ( Arith  x, const var &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

Arith

An arithmetic type

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument and the second argument's value

Definition at line 48 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( double  x, const fvar< T > &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument and the second argument's value

Definition at line 49 of file operator_logical_or.hpp.

template<bool Colwise, bool Rowwise, typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>

optional_broadcast_<as_operation_cl_t<T>, Colwise, Rowwise> stan::math::optional_broadcast ( T &&  a )

inline

Broadcast an expression in specified dimension(s) if the size along that dimension equals 1.

In that case further expressions can use this expression as if it had any size in the broadcast dimension, repeating the values.

Broadcasting evaluates the expression argument multiple times. For performance reasons don't broadcast slow operations. Instead evaluate them in a separate kernel.

Template Parameters

Colwise	whether to broadcast colwise
Rowwise	whether to broadcast rowwise
T	type of input expression

Parameters

a	input expression

Returns

broadcast expression

Definition at line 202 of file optional_broadcast.hpp.

template<typename T , require_rev_col_vector_t< T > * = nullptr>

auto stan::math::ordered_constrain ( const T &  x )

inline

Return an increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters

x	Free vector of scalars

Returns

Increasing ordered vector

Definition at line 25 of file ordered_constrain.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr, require_not_st_var< EigVec > * = nullptr>

plain_type_t<EigVec> stan::math::ordered_constrain ( const EigVec &  x )

inline

Return an increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of the vector

Parameters

x	Free vector of scalars.

Returns

Positive, increasing ordered vector.

Template Parameters

T	Type of scalar.

Definition at line 26 of file ordered_constrain.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>

auto stan::math::ordered_constrain ( const EigVec &  x, value_type_t< EigVec > &  lp  )

inline

Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of the vector

Parameters

x	Free vector of scalars.
lp	Log probability reference.

Returns

Positive, increasing ordered vector.

Definition at line 54 of file ordered_constrain.hpp.

template<typename VarVec , require_var_col_vector_t< VarVec > * = nullptr>

auto stan::math::ordered_constrain	(	const VarVec &	x,
		scalar_type_t< VarVec > &	lp
	)

Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of the vector

Parameters

x	Free vector of scalars.
lp	Log probability reference.

Returns

Positive, increasing ordered vector.

Definition at line 73 of file ordered_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::ordered_constrain ( const T &  x, return_type_t< T > &  lp  )

inline

Return a positive valued, increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Free vector of scalars
[in,out]	lp	log density accumulator

Returns

Positive, increasing ordered vector.

The returned constrained vector will have the same dimensionality as the specified free vector. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	x	Free vector of scalars
[in,out]	lp	log density accumulator

Returns

Positive, increasing ordered vector.

Definition at line 81 of file ordered_constrain.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>

plain_type_t<EigVec> stan::math::ordered_free

(

const EigVec &

y

)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in ordered_constrain(Matrix),

Template Parameters

T	type of elements in the vector

Parameters

y	Vector of positive, ordered scalars.

Returns

Free vector that transforms into the input vector.

Exceptions

std::domain_error

if y is not a vector of positive, ordered scalars.

Definition at line 28 of file ordered_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::ordered_free

(

const T &

x

)

Overload of ordered_free() to untransform each Eigen vector in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase with compile time rows or columns equal to 1.

Parameters

x	The standard vector to untransform.

Definition at line 52 of file ordered_free.hpp.

template<typename T_y , typename T_x , typename T_beta , typename T_cuts >

return_type_t<T_x, T_beta, T_cuts> stan::math::ordered_logistic_glm_lpmf	(	const T_y &	y,
		const T_x &	x,
		const T_beta &	beta,
		const T_cuts &	cuts
	)

Definition at line 208 of file ordered_logistic_glm_lpmf.hpp.

template<typename T_y , typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_logistic_lpmf	(	const T_y &	y,
		const T_loc &	lambda,
		const T_cut &	c
	)

Definition at line 208 of file ordered_logistic_lpmf.hpp.

template<class RNG >

int stan::math::ordered_logistic_rng ( double  eta, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  c, RNG &  rng  )

inline

Definition at line 14 of file ordered_logistic_rng.hpp.

template<typename T_y , typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_probit_lpmf	(	const T_y &	y,
		const T_loc &	lambda,
		const T_cut &	c
	)

Definition at line 104 of file ordered_probit_lpmf.hpp.

template<class RNG >

int stan::math::ordered_probit_rng ( double  eta, const Eigen::VectorXd &  c, RNG &  rng  )

inline

Definition at line 13 of file ordered_probit_rng.hpp.

void stan::math::out_of_range ( const char *  function, int  max, int  index, const char *  msg1 = "", const char *  msg2 = ""  )

inline

Throw an out_of_range exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing out of range. This will allow us to change the behavior for all functions at once. The message is: "<function>: index <index> out of range; expecting index to be between " "1 and <max><msg1><msg2>"

Parameters

function	Name of the function
max	Max
index	Index
msg1	Message to print. Default is "".
msg2	Message to print. Default is "".

Exceptions

std::out_of_range

with message.

Definition at line 27 of file out_of_range.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return Owen's T function applied to the specified arguments.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 25 of file owens_t.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( double  x1, const fvar< T > &  x2  )

inline

Return Owen's T function applied to the specified arguments.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 47 of file owens_t.hpp.

double stan::math::owens_t ( double  h, double  a  )

inline

Return the result of applying Owen's T function to the specified arguments.

Used to compute the cumulative density function for the skew normal distribution.

\[ \mbox{owens\_t}(h, a) = \begin{cases} \mbox{owens\_t}(h, a) & \mbox{if } -\infty\leq h, a \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \mbox{owens\_t}(h, a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial h} = -\frac{1}{2\sqrt{2\pi}} \operatorname{erf}\left(\frac{ha}{\sqrt{2}}\right) \exp\left(-\frac{h^2}{2}\right) \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial a} = \frac{\exp\left(-\frac{1}{2}h^2(1+a^2)\right)}{2\pi (1+a^2)} \]

Parameters

h	First argument
a	Second argument

Returns

Owen's T function applied to the arguments.

Definition at line 58 of file owens_t.hpp.

var stan::math::owens_t ( const var &  h, const var &  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	var parameter.

Returns

The Owen's T function.

Definition at line 64 of file owens_t.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( const fvar< T > &  x1, double  x2  )

inline

Return Owen's T function applied to the specified arguments.

Template Parameters

T	inner type of the fvar

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 66 of file owens_t.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::owens_t ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the owens_t function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

owens_t function applied to the two inputs.

Definition at line 71 of file owens_t.hpp.

var stan::math::owens_t ( const var &  h, double  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	double parameter.

Returns

The Owen's T function.

Definition at line 78 of file owens_t.hpp.

var stan::math::owens_t ( double  h, const var &  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	double parameter.
a	var parameter.

Returns

The Owen's T function.

Definition at line 92 of file owens_t.hpp.

template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_cdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 23 of file pareto_cdf.hpp.

template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lccdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 27 of file pareto_lccdf.hpp.

template<typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lcdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 27 of file pareto_lcdf.hpp.

template<bool propto, typename T_y , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lpdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 27 of file pareto_lpdf.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lpdf ( const T_y &  y, const T_scale &  y_min, const T_shape &  alpha  )

inline

Definition at line 96 of file pareto_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 24 of file pareto_type_2_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 24 of file pareto_type_2_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 24 of file pareto_type_2_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 27 of file pareto_type_2_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  lambda, const T_shape &  alpha  )

inline

Definition at line 112 of file pareto_type_2_lpdf.hpp.

template<typename T , typename F >

void stan::math::partial_derivative	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		int	n,
		T &	fx,
		T &	dfx_dxn
	)

Return the partial derivative of the specified multivariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

	f	Function
[in]	x	Argument vector
[in]	n	Index of argument with which to take derivative
[out]	fx	Value of function applied to argument
[out]	dfx_dxn	Value of partial derivative

Definition at line 25 of file partial_derivative.hpp.

template<typename T >

fvar<T> stan::math::Phi ( const fvar< T > &  x )

inline

Definition at line 14 of file Phi.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::Phi ( const var_value< T > &  A )

inline

Returns the elementwise Phi() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise Phi() of the input.

Definition at line 20 of file Phi.hpp.

double stan::math::Phi ( double  x )

inline

The unit normal cumulative distribution function.

The return value for a specified input is the probability that a random unit normal variate is less than or equal to the specified value, defined by

\(\Phi(x) = \int_{-\infty}^x \mbox{\sf Norm}(x|0, 1) \ dx\)

This function can be used to implement the inverse link function for probit regression.

Phi will underflow to 0 below -37.5 and overflow to 1 above 8

Parameters

x

Argument.

Returns

Probability random sample is less than or equal to argument.

Definition at line 32 of file Phi.hpp.

var stan::math::Phi ( const var &  a )

inline

The unit normal cumulative density function for variables (stan).

See Phi() for the double-based version.

The derivative is the unit normal density function,

\(\frac{d}{dx} \Phi(x) = \mbox{\sf Norm}(x|0, 1) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2)\).

\[ \mbox{Phi}(x) = \begin{cases} 0 & \mbox{if } x < -37.5 \\ \Phi(x) & \mbox{if } -37.5 \leq x \leq 8.25 \\ 1 & \mbox{if } x > 8.25 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } x < -27.5 \\ \frac{\partial\, \Phi(x)}{\partial x} & \mbox{if } -27.5 \leq x \leq 27.5 \\ 0 & \mbox{if } x > 27.5 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{0}^{x} e^{-t^2/2} dt \]

\[ \frac{\partial \, \Phi(x)}{\partial x} = \frac{e^{-x^2/2}}{\sqrt{2\pi}} \]

Parameters

a	Variable argument.

Returns

The unit normal cdf evaluated at the specified argument.

Definition at line 53 of file Phi.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::Phi ( const T &  a )

inline

Elementwise unit normal cumulative density function for varmat types.

Template Parameters

T	a var_value with inner Eigen type

Parameters

a

input

Returns

The vectorized unit normal cdf

Definition at line 67 of file Phi.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::Phi ( const T &  x )

inline

Vectorized version of Phi().

Template Parameters

T	type of container

Parameters

x

container

Returns

Unit normal CDF of each value in x.

Definition at line 70 of file Phi.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::Phi_approx ( const var_value< T > &  A )

inline

Returns the elementwise Phi_approx() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise Phi_approx() of the input.

Definition at line 20 of file Phi_approx.hpp.

template<typename T >

fvar<T> stan::math::Phi_approx ( const fvar< T > &  x )

inline

Return an approximation of the unit normal cumulative distribution function (CDF).

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

argument

Returns

approximate probability random sample is less than or equal to argument

Definition at line 23 of file Phi_approx.hpp.

double stan::math::Phi_approx ( double  x )

inline

Return an approximation of the unit normal CDF.

http://www.jiem.org/index.php/jiem/article/download/60/27

This function can be used to implement the inverse link function for probit regression.

Parameters

x

Argument.

Returns

Probability random sample is less than or equal to argument.

Definition at line 23 of file Phi_approx.hpp.

double stan::math::Phi_approx ( int  x )

inline

Return an approximation of the unit normal CDF.

Parameters

x

argument.

Returns

approximate probability random sample is less than or equal to argument.

Definition at line 35 of file Phi_approx.hpp.

var stan::math::Phi_approx ( const var &  a )

inline

Approximation of the unit normal CDF for variables (stan).

http://www.jiem.org/index.php/jiem/article/download/60/27

\[ \mbox{Phi\_approx}(x) = \begin{cases} \Phi_{\mbox{\footnotesize approx}}(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi\_approx}(x)}{\partial x} = \begin{cases} \frac{\partial\, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi_{\mbox{\footnotesize approx}}(x) = \mbox{logit}^{-1}(0.07056 \, x^3 + 1.5976 \, x) \]

\[ \frac{\partial \, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} = -\Phi_{\mbox{\footnotesize approx}}^2(x) e^{-0.07056x^3 - 1.5976x}(-0.21168x^2-1.5976) \]

Parameters

a	Variable argument.

Returns

The corresponding unit normal cdf approximation.

Definition at line 46 of file Phi_approx.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::Phi_approx ( const T &  a )

inline

Definition at line 55 of file Phi_approx.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::Phi_approx ( const T &  x )

inline

Return the elementwise application of Phi_approx() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise Phi_approx of container elements

Definition at line 69 of file Phi_approx.hpp.

static constexpr double stan::math::pi ( )

static

Return the value of pi.

Returns

Pi.

Definition at line 36 of file constants.hpp.

template<typename T >

T stan::math::plus ( T &&  x )

inline

Returns the unary plus of the input.

Template Parameters

T	Type of input.

Parameters

x

input.

Returns

result of unary plus of the input.

Definition at line 17 of file plus.hpp.

template<typename T_y , typename T_theta >

return_type_t<T_theta> stan::math::poisson_binomial_cdf	(	const T_y &	y,
		const T_theta &	theta
	)

Definition at line 66 of file poisson_binomial_cdf.hpp.

template<typename T_y , typename T_theta >

return_type_t<T_theta> stan::math::poisson_binomial_lccdf	(	const T_y &	y,
		const T_theta &	theta
	)

Definition at line 77 of file poisson_binomial_lccdf.hpp.

template<typename T_y , typename T_theta >

return_type_t<T_theta> stan::math::poisson_binomial_lcdf	(	const T_y &	y,
		const T_theta &	theta
	)

Definition at line 67 of file poisson_binomial_lcdf.hpp.

template<typename T_theta , typename T_scalar = scalar_type_t<T_theta>, require_eigen_vector_t< T_theta > * = nullptr>

plain_type_t<T_theta> stan::math::poisson_binomial_log_probs	(	int	y,
		const T_theta &	theta
	)

Returns the last row of the log probability matrix of the Poisson-Binomial distribution given the number of successes and a vector of success probabilities.

Template Parameters

T_theta

template expression

Parameters

y	numbers of successes
theta	N-dimensional vector of success probabilities for each trial

Returns

the last row of the computed log probability matrix

Definition at line 26 of file poisson_binomial_log_probs.hpp.

template<typename T_y , typename T_theta , require_vt_integral< T_y > * = nullptr>

auto stan::math::poisson_binomial_log_probs	(	const T_y &	y,
		const T_theta &	theta
	)

Definition at line 56 of file poisson_binomial_log_probs.hpp.

template<typename T_y , typename T_theta >

return_type_t<T_theta> stan::math::poisson_binomial_lpmf	(	const T_y &	y,
		const T_theta &	theta
	)

Definition at line 60 of file poisson_binomial_lpmf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_cdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 27 of file poisson_cdf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lccdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 28 of file poisson_lccdf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lcdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 28 of file poisson_lcdf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta >

return_type_t<T_x, T_alpha, T_beta> stan::math::poisson_log_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 166 of file poisson_log_glm_lpmf.hpp.

template<bool propto, typename T_n , typename T_log_rate , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_log_rate > * = nullptr>

return_type_t<T_log_rate> stan::math::poisson_log_lpmf	(	const T_n &	n,
		const T_log_rate &	alpha
	)

Definition at line 29 of file poisson_log_lpmf.hpp.

template<typename T_n , typename T_log_rate >

return_type_t<T_log_rate> stan::math::poisson_log_lpmf ( const T_n &  n, const T_log_rate &  alpha  )

inline

Definition at line 90 of file poisson_log_lpmf.hpp.

template<bool propto, typename T_n , typename T_rate , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_n, T_rate > * = nullptr>

return_type_t<T_rate> stan::math::poisson_lpmf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 29 of file poisson_lpmf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lpmf ( const T_n &  n, const T_rate &  lambda  )

inline

Definition at line 85 of file poisson_lpmf.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::polar ( const fvar< T > &  r, const fvar< T > &  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Template Parameters

T	autodiff value type

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 22 of file polar.hpp.

std::complex<var> stan::math::polar ( const var &  r, const var &  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 26 of file polar.hpp.

template<typename T , typename U >

std::complex<fvar<T> > stan::math::polar ( const fvar< T > &  r, U  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Template Parameters

T	autodiff value type for magnitude
U	arithmetic type for phase angle

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 36 of file polar.hpp.

std::complex<double> stan::math::polar ( double  r, double  theta  )

inline

Returns the complex number with specified magnitude and phase angle.

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 39 of file polar.hpp.

template<typename T >

std::complex<var> stan::math::polar ( T  r, const var &  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Template Parameters

T	arithmetic type of magnitude

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 39 of file polar.hpp.

std::complex<double> stan::math::polar ( double  r, int  theta  )

inline

Definition at line 42 of file polar.hpp.

std::complex<double> stan::math::polar ( int  r, double  theta  )

inline

Definition at line 45 of file polar.hpp.

std::complex<double> stan::math::polar ( int  r, int  theta  )

inline

Definition at line 48 of file polar.hpp.

template<typename T , typename U >

std::complex<fvar<T> > stan::math::polar ( U  r, const fvar< T > &  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Template Parameters

T	autodiff value type for phase angle +* *
U	arithmetic type for magnitude

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 50 of file polar.hpp.

template<typename T >

std::complex<var> stan::math::polar ( const var &  r, T  theta  )

inline

Returns complex number with specified magnitude and phase angle.

Template Parameters

T	arithmetic type of phase angle

Parameters

[in]	r	magnitude
[in]	theta	phase angle

Returns

complex number with magnitude and phase angle

Definition at line 52 of file polar.hpp.

template<typename T >

auto stan::math::positive_constrain ( const T &  x )

inline

Return the positive value for the specified unconstrained input.

The transform applied is

\(f(x) = \exp(x)\).

Parameters

x	Arbitrary input scalar or container.

Returns

Input transformed to be positive.

Definition at line 23 of file positive_constrain.hpp.

template<typename T , typename S >

auto stan::math::positive_constrain ( const T &  x, S &  lp  )

inline

Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant.

See positive_constrain(T) for details of the transform. The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \mbox{exp}(x) | = \log | \mbox{exp}(x) | = x\).

Template Parameters

T	type of unconstrained value

Parameters

x	unconstrained value or container
lp	log density reference.

Returns

positive constrained version of unconstrained value(s)

Definition at line 44 of file positive_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::positive_constrain ( const T &  x, return_type_t< T > &  lp  )

inline

Return the positive value for the specified unconstrained input.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

	x	unconstrained value or container
[in,out]	lp	log density accumulator

Returns

positive constrained version of unconstrained value(s)

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

	x	unconstrained value or container
[in,out]	lp	log density accumulator

Returns

positive constrained version of unconstrained value(s)

Definition at line 65 of file positive_constrain.hpp.

template<typename T >

T stan::math::positive_free ( const T &  y )

inline

Return the unconstrained value corresponding to the specified positive-constrained value.

The transform is the inverse of the transform \(f\) applied by positive_constrain(T), namely

\(f^{-1}(x) = \log(x)\).

The input is validated using check_positive().

Parameters

y	Input scalar.

Returns

Unconstrained value that produces the input when constrained.

Template Parameters

T	Type of scalar.

Exceptions

std::domain_error

if the variable is negative

Definition at line 29 of file positive_free.hpp.

static constexpr double stan::math::positive_infinity ( )

inlinestatic

Return positive infinity.

Returns

Positive infinity.

Definition at line 199 of file constants.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr, require_not_st_var< EigVec > * = nullptr>

auto stan::math::positive_ordered_constrain ( const EigVec &  x )

inline

Return an increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.

Returns

Positive, increasing ordered vector.

Definition at line 24 of file positive_ordered_constrain.hpp.

template<typename T , require_rev_col_vector_t< T > * = nullptr>

auto stan::math::positive_ordered_constrain ( const T &  x )

inline

Return an increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters

x	Free vector of scalars

Returns

Positive, increasing ordered vector

Definition at line 25 of file positive_ordered_constrain.hpp.

template<typename Vec , require_col_vector_t< Vec > * = nullptr>

auto stan::math::positive_ordered_constrain ( const Vec &  x, return_type_t< Vec > &  lp  )

inline

Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.
lp	Log probability reference.

Returns

Positive, increasing ordered vector.

Definition at line 52 of file positive_ordered_constrain.hpp.

template<bool Jacobian, typename Vec , require_not_std_vector_t< Vec > * = nullptr>

auto stan::math::positive_ordered_constrain ( const Vec &  x, return_type_t< Vec > &  lp  )

inline

Return a positive valued, increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
Vec	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase

Parameters

	x	Free vector of scalars
[in,out]	lp	log density accumulato

Returns

Positive, increasing ordered vector

Definition at line 76 of file positive_ordered_constrain.hpp.

template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::positive_ordered_constrain ( const T &  x, return_type_t< T > &  lp  )

inline

Return a positive valued, increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector. If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
Vec	A standard vector with inner type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase

Parameters

	x	Free vector of scalars
[in,out]	lp	log density accumulato

Returns

Positive, increasing ordered vector

Definition at line 103 of file positive_ordered_constrain.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>

auto stan::math::positive_ordered_free

(

const EigVec &

y

)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in positive_ordered_constrain(Matrix),

Template Parameters

T	type of elements in the vector

Parameters

y	Vector of positive, ordered scalars.

Returns

Free vector that transforms into the input vector.

Exceptions

std::domain_error

if y is not a vector of positive, ordered scalars.

Definition at line 28 of file positive_ordered_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::positive_ordered_free

(

const T &

x

)

Overload of positive_ordered_free() to untransform each Eigen vector in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase with compile time rows or columns equal to 1.

Parameters

x	The standard vector to untransform.

Definition at line 52 of file positive_ordered_free.hpp.

template<typename T >

fvar<T> stan::math::pow ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 19 of file pow.hpp.

template<typename T , typename U , typename = require_arithmetic_t<U>>

fvar<T> stan::math::pow ( U  x1, const fvar< T > &  x2  )

inline

Definition at line 28 of file pow.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

var_value<matrix_cl<double> > stan::math::pow ( T_a &&  a, T_b &&  b  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T_a	type of first expression of the base
T_b	type of second expression of the exponent

Parameters

a	first expression of base
b	second expression of exponent

Returns

base raised to the power of the exponent

Definition at line 33 of file pow.hpp.

template<typename T , typename U , typename = require_arithmetic_t<U>>

fvar<T> stan::math::pow ( const fvar< T > &  x1, U  x2  )

inline

Definition at line 36 of file pow.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::pow ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the pow function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

pow function applied to the two inputs.

Definition at line 42 of file pow.hpp.

template<typename V >

std::complex<fvar<V> > stan::math::pow ( const std::complex< fvar< V >> &  x, const std::complex< fvar< V >> &  y  )

inline

Return the first argument raised to the power of the second argument.

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 81 of file pow.hpp.

template<typename V , typename T , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( const std::complex< fvar< V >> &  x, const std::complex< T > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 96 of file pow.hpp.

template<typename V >

std::complex<fvar<V> > stan::math::pow ( const std::complex< fvar< V >> &  x, const fvar< V > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 110 of file pow.hpp.

var stan::math::pow ( const var &  base, const var &  exponent  )

inline

Return the base raised to the power of the exponent (cmath).

The partial derivatives are

\(\frac{\partial}{\partial x} \mbox{pow}(x, y) = y x^{y-1}\), and

\(\frac{\partial}{\partial y} \mbox{pow}(x, y) = x^y \ \log x\).

\[ \mbox{pow}(x, y) = \begin{cases} x^y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial x} = \begin{cases} yx^{y-1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial y} = \begin{cases} x^y\ln x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

base	Base variable.
exponent	Exponent variable.

Returns

Base raised to the exponent.

Definition at line 119 of file pow.hpp.

template<typename V , typename T , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( const std::complex< fvar< V >> &  x, const T &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 125 of file pow.hpp.

template<typename V , typename T , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( const std::complex< T > &  x, const std::complex< fvar< V >> &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 139 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

var stan::math::pow ( const var &  base, T  exponent  )

inline

Return the base variable raised to the power of the exponent scalar (cmath).

The derivative for the variable is

\(\frac{d}{dx} \mbox{pow}(x, c) = c x^{c-1}\).

The template parameters are coded as they are so that arithmetic types will not be promoted into the var slots.

Template Parameters

T	arithmetic type

Parameters

base	Base variable.
exponent	Exponent scalar.

Returns

Base raised to the exponent.

Definition at line 140 of file pow.hpp.

template<typename V , typename T , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( const std::complex< T > &  x, const fvar< V > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 154 of file pow.hpp.

template<typename V >

std::complex<fvar<V> > stan::math::pow ( const fvar< V > &  x, const std::complex< fvar< V >> &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 167 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

var stan::math::pow ( T  base, const var &  exponent  )

inline

Return the base scalar raised to the power of the exponent variable (cmath).

The derivative for the variable is

\(\frac{d}{d y} \mbox{pow}(c, y) = c^y \log c \).

The template parameters are coded as they are so that arithmetic types will not be promoted into the var slots.

Template Parameters

T	arithmetic type

Parameters

base	Base scalar.
exponent	Exponent variable.

Returns

Base raised to the exponent.

Definition at line 176 of file pow.hpp.

template<typename V , typename T , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( const fvar< V > &  x, const std::complex< T > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 182 of file pow.hpp.

template<typename T , typename V , typename = require_arithmetic_t<T>>

std::complex<fvar<V> > stan::math::pow ( T  x, const std::complex< fvar< V >> &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

V	autodiff value type
T	real type (fvar<V> or arithmetic)

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 196 of file pow.hpp.

std::complex<var> stan::math::pow ( const std::complex< var > &  x, const std::complex< var > &  y  )

inline

Return the first argument raised to the power of the second argument.

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 199 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( const std::complex< var > &  x, const std::complex< T >  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 213 of file pow.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::pow ( const std::complex< fvar< T >> &  x, int  y  )

inline

Return the first argument raised to the power of the second argument.

Note: this overload is required because gcc still provides the C++99 template function pow(complex<T>, int), which introduces an ambiguity.

Template Parameters

T	autodiff value type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 213 of file pow.hpp.

std::complex<var> stan::math::pow ( const std::complex< var > &  x, const var &  y  )

inline

Return the first argument raised to the power of the second argument.

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 225 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( const std::complex< var > &  x, T  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 238 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( std::complex< T >  x, const std::complex< var > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 251 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( std::complex< T >  x, const var &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 264 of file pow.hpp.

std::complex<var> stan::math::pow ( const var &  x, const std::complex< var > &  y  )

inline

Return the first argument raised to the power of the second argument.

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 275 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( const var &  x, std::complex< T >  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 288 of file pow.hpp.

template<typename T , typename = require_arithmetic_t<T>>

std::complex<var> stan::math::pow ( T  x, const std::complex< var > &  y  )

inline

Return the first argument raised to the power of the second argument.

Template Parameters

T	arithmetic type

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 301 of file pow.hpp.

std::complex<var> stan::math::pow ( const std::complex< var > &  x, int  y  )

inline

Return the first argument raised to the power of the second argument.

Note: this overload is required because gcc still provides the C++99 template function pow(complex<T>, int), which introduces an ambiguity.

Parameters

x	first argument
y	second argument

Returns

first argument to the power of the second argument

Definition at line 316 of file pow.hpp.

template<typename Arith , typename VecVar , typename VecArith , typename... ContainerOperands, typename... ContainerGradients>

var stan::math::precomputed_gradients ( Arith  value, const VecVar &  operands, const VecArith &  gradients, const std::tuple< ContainerOperands... > &  container_operands = std::tuple<>(), const std::tuple< ContainerGradients... > &  container_gradients = std::tuple<>()  )

inline

This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands.

Template Parameters

Arith	An arithmetic type
VecVar	A vector of vars
VecArith	A vector of arithmetic types
ContainerOperands	tuple of any container operands (var_value containing Eigen types)
ContainerGradients	tupleof any container gradients (Eigen types)

Parameters

[in]	value	The value of the resulting dependent variable.
[in]	operands	operands.
[in]	gradients	vector of partial derivatives of result with respect to operands.
	container_operands	any container operands
	container_gradients	any container gradients

Returns

An autodiff variable that uses the precomputed gradients provided.

Definition at line 213 of file precomputed_gradients.hpp.

double stan::math::primitive_value ( const var &  v )

inline

Return the primitive double value for the specified autodiff variable.

Parameters

v	input variable.

Returns

value of input.

Definition at line 18 of file primitive_value.hpp.

template<typename T >

double stan::math::primitive_value ( const fvar< T > &  v )

inline

Return the primitive value of the specified forward-mode autodiff variable.

This function applies recursively to higher-order autodiff types to return a primitive double value.

Template Parameters

T	inner type of the fvar

Parameters

v	input variable.

Returns

primitive value of input.

Definition at line 21 of file primitive_value.hpp.

template<typename T >

std::enable_if<std::is_arithmetic<T>::value, T>::type stan::math::primitive_value ( T  x )

inline

Return the value of the specified arithmetic argument unmodified with its own declared type.

This template function can only be instantiated with arithmetic types as defined by std library's is_arithmetic trait metaprogram.

This function differs from value_of in that it does not cast all return types to double.

Template Parameters

T	type of arithmetic input.

Parameters

x

input.

Returns

input unmodified.

Definition at line 28 of file primitive_value.hpp.

template<typename T >

std::enable_if<!std::is_arithmetic<T>::value, double>::type stan::math::primitive_value ( const T &  x )

inline

Return the primitive value of the specified argument.

This implementation only applies to non-arithmetic types as defined by std library's is_arithmetic trait metaprogram.

Template Parameters

T	type of non-arithmetic input.

Parameters

x

input.

Returns

value of input.

Definition at line 44 of file primitive_value.hpp.

template<int N>

void stan::math::print_mat_size ( std::ostream &  o )

inline

Helper function to return the matrix size as either "dynamic" or "1".

Template Parameters

N	Eigen matrix size specification

Parameters

o	output stream

Definition at line 22 of file assign.hpp.

void stan::math::print_stack ( std::ostream &  o )

inline

Prints the autodiff variable stack.

This function is used for debugging purposes.

Only works if all members of stack are vari* as it casts to vari*.

Parameters

o	ostream to modify

Definition at line 20 of file print_stack.hpp.

template<typename T >

T stan::math::prob_constrain ( const T &  x )

inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar.

The transform is the inverse logit,

\(f(x) = \mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(x)}\).

Template Parameters

T	type of scalar

Parameters

[in]

x

unconstrained value

Returns

result constrained to fall in (0, 1)

Definition at line 26 of file prob_constrain.hpp.

template<typename T >

T stan::math::prob_constrain ( const T &  x, T &  lp  )

inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is as defined for prob_constrain(T). The log absolute Jacobian determinant is

The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \mbox{logit}^{-1}(x) |\)

\(\log ((\mbox{logit}^{-1}(x)) (1 - \mbox{logit}^{-1}(x))\)

\(\log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))\).

Template Parameters

T	type of scalar

Parameters

[in]	x	unconstrained value
[in,out]	lp	log density

Returns

result constrained to fall in (0, 1)

Definition at line 51 of file prob_constrain.hpp.

template<bool Jacobian, typename T >

auto stan::math::prob_constrain ( const T &  x, T &  lp  )

inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	type of scalar

Parameters

[in]	x	unconstrained value
[in,out]	lp	log density accumulator

Returns

result constrained to fall in (0, 1)

Definition at line 73 of file prob_constrain.hpp.

template<typename T >

T stan::math::prob_free ( const T &  y )

inline

Return the free scalar that when transformed to a probability produces the specified scalar.

The function that reverses the constraining transform specified in prob_constrain(T) is the logit function,

\(f^{-1}(y) = \mbox{logit}(y) = \frac{1 - y}{y}\).

Template Parameters

T	type of constrained value

Parameters

y	constrained value

Returns

corresponding unconstrained value

Exceptions

std::domain_error

if y is not in (0, 1)

Definition at line 27 of file prob_free.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T stan::math::prod

(

const T &

v

)

Returns the product of given scalar.

This is a no-op.

Template Parameters

T	type of the scalar

Parameters

v	specified scalar

Returns

the scalar

Definition at line 19 of file prod.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

value_type_t<T> stan::math::prod

(

const T &

m

)

Calculates product of given kernel generator expression elements.

Template Parameters

T	type of the expression

Parameters

m	expression to calcualte product of

Returns

product of given expression

Definition at line 21 of file prod.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::prod ( const var_value< T > &  x )

inline

Returns the prod of the coefficients of the specified matrix on the OpenCL device.

Parameters

x	Specified var_value containing a matrix.

Returns

prod of coefficients of matrix.

Definition at line 22 of file prod.hpp.

template<typename T >

T stan::math::prod ( const std::vector< T > &  v )

inline

Returns the product of the coefficients of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Product of coefficients of vector.

Definition at line 32 of file prod.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

value_type_t<EigMat> stan::math::prod ( const EigMat &  v )

inline

Returns the product of the coefficients of the specified column vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Product of coefficients of vector.

Definition at line 49 of file prod.hpp.

std::complex<var> stan::math::proj ( const std::complex< var > &  z )

inline

Return the projection of the complex argument onto the Riemann sphere.

Parameters

[in]

z

argument

Returns

projection of the argument onto the Riemann sphere

Definition at line 20 of file proj.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::proj ( const std::complex< fvar< T >> &  z )

inline

Return the projection of the complex argument onto the Riemann sphere.

Template Parameters

T	value type of autodiff variable

Parameters

[in]

z

argument

Returns

projection of the argument onto the Riemann sphere

Definition at line 20 of file proj.hpp.

template<typename T , typename S >

promote_scalar_type<T, S>::type stan::math::promote_scalar

(

const S &

x

)

This is the top-level function to call to promote the scalar types of an input of type S to type T.

Template Parameters

T	scalar type of output.
S	input type.

Parameters

x	input vector.

Returns

input vector with scalars promoted to type T.

Definition at line 63 of file promote_scalar.hpp.

template<typename T >

Eigen::Matrix<T, -1, -1> stan::math::pseudo_eigenvalues

(

const Eigen::Matrix< T,-1,-1 > &

m

)

Definition at line 11 of file pseudo_eigenvalues.hpp.

template<typename T >

Eigen::Matrix<T, -1, -1> stan::math::pseudo_eigenvectors

(

const Eigen::Matrix< T,-1,-1 > &

m

)

Definition at line 11 of file pseudo_eigenvectors.hpp.

template<bool need_Q = true>

void stan::math::qr_decomposition_cl	(	const matrix_cl< double > &	A,
		matrix_cl< double > &	Q,
		matrix_cl< double > &	R,
		int	r = 100
	)

Calculates QR decomposition of A using the block Householder algorithm.

The implementation is based on the article: need_Q whether Q needs to be calculated (default true) A matrix to factorize Q out the orthonormal matrix R out the lower triangular matrix r Block size. Optimal value depends on the hardware.

Definition at line 34 of file qr_decomposition.hpp.

template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>

matrix_cl<double> stan::math::qr_Q ( T_m &&  m )

inline

Returns the orthogonal factor of the fat QR decomposition.

Template Parameters

T_m	type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 20 of file qr_Q.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_Q

(

const EigMat &

m

)

Returns the orthogonal factor of the fat QR decomposition.

Template Parameters

EigMat

type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 20 of file qr_Q.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_R

(

const EigMat &

m

)

Returns the upper triangular factor of the fat QR decomposition.

Template Parameters

EigMat

type of the matrix

Parameters

m

Matrix.

Returns

Upper triangular matrix with maximal rows

Definition at line 19 of file qr_R.hpp.

template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>

matrix_cl<double> stan::math::qr_R ( T_m &&  m )

inline

Returns the orthogonal factor of the fat QR decomposition.

Template Parameters

T_m	type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 20 of file qr_R.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_thin_Q

(

const EigMat &

m

)

Returns the orthogonal factor of the thin QR decomposition.

Template Parameters

EigMat

type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with minimal columns

Definition at line 19 of file qr_thin_Q.hpp.

template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>

matrix_cl<double> stan::math::qr_thin_Q ( T_m &&  m )

inline

Returns the orthogonal factor of the fat QR decomposition.

Template Parameters

T_m	type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 20 of file qr_thin_Q.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_thin_R

(

const EigMat &

m

)

Returns the upper triangular factor of the thin QR decomposition.

Template Parameters

EigMat

type of the matrix

Parameters

m

Matrix.

Returns

Upper triangular matrix with minimal rows

Definition at line 19 of file qr_thin_R.hpp.

template<typename T_m , require_all_kernel_expressions_and_none_scalar_t< T_m > * = nullptr>

matrix_cl<double> stan::math::qr_thin_R ( T_m &&  m )

inline

Returns the orthogonal factor of the thin QR decomposition.

Template Parameters

T_m	type of the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 20 of file qr_thin_R.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr, require_all_vt_arithmetic< EigMat1, EigMat2 > * = nullptr>

auto stan::math::quad_form ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return the quadratic form \( B^T A B \).

Symmetry of the resulting matrix is not guaranteed due to numerical precision.

Template Parameters

EigMat1	type of the first (square) matrix
EigMat2	type of the second matrix

Parameters

A	square matrix
B	second matrix

Returns

The quadratic form, which is a symmetric matrix.

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 31 of file quad_form.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>

promote_scalar_t<return_type_t<EigMat1, EigMat2>, EigMat2> stan::math::quad_form ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return the quadratic form \( B^T A B \).

Symmetry of the resulting matrix is not guaranteed due to numerical precision.

Template Parameters

EigMat1	type of the first (square) matrix
EigMat2	type of the second matrix

Parameters

A	square matrix
B	second matrix

Returns

The quadratic form, which is a symmetric matrix of size Cb.

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 31 of file quad_form.hpp.

template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_fvar< EigMat, ColVec > * = nullptr>

return_type_t<EigMat, ColVec> stan::math::quad_form ( const EigMat &  A, const ColVec &  B  )

inline

Return the quadratic form \( B^T A B \).

Template Parameters

EigMat	type of the matrix
ColVec	type of the vector

Parameters

A	square matrix
B	vector

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 54 of file quad_form.hpp.

template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_vt_same< EigMat, ColVec > * = nullptr, require_all_vt_arithmetic< EigMat, ColVec > * = nullptr>

value_type_t<EigMat> stan::math::quad_form ( const EigMat &  A, const ColVec &  B  )

inline

Return the quadratic form \( B^T A B \).

Template Parameters

EigMat	type of the matrix
ColVec	type of the vector

Parameters

A	square matrix
B	vector

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 55 of file quad_form.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_var< EigMat1, EigMat2 > * = nullptr>

promote_scalar_t<var, EigMat2> stan::math::quad_form ( const EigMat1 &  A, const EigMat2 &  B, bool  symmetric = false  )

inline

Return the quadratic form \( B^T A B \).

Symmetry of the resulting matrix is not guaranteed due to numerical precision.

Template Parameters

EigMat1	type of the first (square) matrix
EigMat2	type of the second matrix

Parameters

A	square matrix
B	second matrix
symmetric	indicates whether the output should be made symmetric

Returns

The quadratic form, which is a symmetric matrix.

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 244 of file quad_form.hpp.

template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_var< EigMat, ColVec > * = nullptr>

var stan::math::quad_form ( const EigMat &  A, const ColVec &  B, bool  symmetric = false  )

inline

Return the quadratic form \( B^T A B \).

Template Parameters

EigMat	type of the matrix
ColVec	type of the vector

Parameters

A	square matrix
B	vector
symmetric	indicates whether the output should be made symmetric

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 274 of file quad_form.hpp.

template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_not_col_vector_t< Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>

auto stan::math::quad_form ( const Mat1 &  A, const Mat2 &  B, bool  symmetric = false  )

inline

Return the quadratic form \( B^T A B \).

Symmetry of the resulting matrix is not guaranteed due to numerical precision.

This overload handles arguments where one of Mat1 or Mat2 are var_value<T> where T is an Eigen type. The other type can also be a var_value or it can be a matrix type that inherits from EigenBase

Template Parameters

Mat1	type of the first (square) matrix
Mat2	type of the second matrix

Parameters

A	square matrix
B	second matrix
symmetric	indicates whether the output should be made symmetric

Returns

The quadratic form, which is a symmetric matrix.

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 311 of file quad_form.hpp.

template<typename Mat , typename Vec , require_matrix_t< Mat > * = nullptr, require_col_vector_t< Vec > * = nullptr, require_any_var_matrix_t< Mat, Vec > * = nullptr>

var stan::math::quad_form ( const Mat &  A, const Vec &  B, bool  symmetric = false  )

inline

Return the quadratic form \( B^T A B \).

This overload handles arguments where one of Mat or Vec are var_value<T> where T is an Eigen type. The other type can also be a var_value, or it can be a type that inherits from EigenBase

Template Parameters

Mat	type of the matrix
Vec	type of the vector

Parameters

A	square matrix
B	vector
symmetric	indicates whether the output should be made symmetric

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not square, or if A cannot be multiplied by B

Definition at line 336 of file quad_form.hpp.

template<typename EigMat , typename EigVec , require_eigen_t< EigMat > * = nullptr, require_eigen_vector_t< EigVec > * = nullptr>

auto stan::math::quad_form_diag ( const EigMat &  mat, const EigVec &  vec  )

inline

Definition at line 13 of file quad_form_diag.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>

promote_scalar_t<return_type_t<EigMat1, EigMat2>, EigMat2> stan::math::quad_form_sym ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return the quadratic form \( B^T A B \) of a symmetric matrix.

Symmetry of the resulting matrix is guaranteed.

Template Parameters

EigMat1	type of the first (symmetric) matrix
EigMat2	type of the second matrix

Parameters

A	symmetric matrix
B	second matrix

Returns

The quadratic form, which is a symmetric matrix of size CB.

Exceptions

std::invalid_argument

if A is not symmetric, or if A cannot be multiplied by B

Definition at line 30 of file quad_form_sym.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_not_eigen_col_vector_t< EigMat2 > * = nullptr, require_vt_same< EigMat1, EigMat2 > * = nullptr, require_all_vt_arithmetic< EigMat1, EigMat2 > * = nullptr>

plain_type_t<EigMat2> stan::math::quad_form_sym ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return the quadratic form \( B^T A B \) of a symmetric matrix.

Symmetry of the resulting matrix is guaranteed.

Template Parameters

EigMat1	type of the first (symmetric) matrix
EigMat2	type of the second matrix

Parameters

A	symmetric matrix
B	second matrix

Returns

The quadratic form, which is a symmetric matrix.

Exceptions

std::invalid_argument

if A is not symmetric, or if A cannot be multiplied by B

Definition at line 30 of file quad_form_sym.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_vt_var< EigMat1, EigMat2 > * = nullptr>

auto stan::math::quad_form_sym ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Return the quadratic form \( B^T A B \) of a symmetric matrix.

Symmetry of the resulting matrix is guaranteed.

Template Parameters

EigMat1	type of the first (symmetric) matrix
EigMat2	type of the second matrix

Parameters

A	symmetric matrix
B	second matrix

Returns

The quadratic form, which is a symmetric matrix of size Cb. If B is a column vector returns a scalar.

Exceptions

std::invalid_argument

if A is not symmetric, or if A cannot be multiplied by B

Definition at line 33 of file quad_form_sym.hpp.

template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_vt_same< EigMat, ColVec > * = nullptr, require_all_vt_arithmetic< EigMat, ColVec > * = nullptr>

value_type_t<EigMat> stan::math::quad_form_sym ( const EigMat &  A, const ColVec &  B  )

inline

Return the quadratic form \( B^T A B \) of a symmetric matrix.

Template Parameters

EigMat	type of the (symmetric) matrix
ColVec	type of the vector

Parameters

A	symmetric matrix
B	vector

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not symmetric, or if A cannot be multiplied by B

Definition at line 56 of file quad_form_sym.hpp.

template<typename EigMat , typename ColVec , require_eigen_t< EigMat > * = nullptr, require_eigen_col_vector_t< ColVec > * = nullptr, require_any_vt_fvar< EigMat, ColVec > * = nullptr>

return_type_t<EigMat, ColVec> stan::math::quad_form_sym ( const EigMat &  A, const ColVec &  B  )

inline

Return the quadratic form \( B^T A B \) of a symmetric matrix.

Template Parameters

EigMat	type of the (symmetric) matrix
ColVec	type of the vector

Parameters

A	symmetric matrix
B	vector

Returns

The quadratic form (a scalar).

Exceptions

std::invalid_argument

if A is not symmetric, or if A cannot be multiplied by B

Definition at line 56 of file quad_form_sym.hpp.

template<typename T , require_vector_t< T > * = nullptr, require_vector_vt< std::is_arithmetic, T > * = nullptr>

double stan::math::quantile ( const T &  samples_vec, const double  p  )

inline

Return sample quantiles corresponding to the given probabilities.

The smallest observation corresponds to a probability of 0 and the largest to a probability of 1.

Implements algorithm 7 from Hyndman, R. J. and Fan, Y., Sample quantiles in Statistical Packages (R's default quantile function)

Template Parameters

T	An Eigen type with either fixed rows or columns at compile time, or std::vector<double>

Parameters

samples_vec	Numeric vector whose sample quantiles are wanted
p	Probability with value between 0 and 1.

Returns

Sample quantile.

Exceptions

std::invalid_argument	If any element of samples_vec is NaN, or size 0.
std::domain_error	If p<0 or p>1.

Definition at line 32 of file quantile.hpp.

template<typename T , typename Tp , require_all_vector_t< T, Tp > * = nullptr, require_vector_vt< std::is_arithmetic, T > * = nullptr, require_std_vector_vt< std::is_arithmetic, Tp > * = nullptr>

std::vector<double> stan::math::quantile ( const T &  samples_vec, const Tp &  ps  )

inline

Return sample quantiles corresponding to the given probabilities.

The smallest observation corresponds to a probability of 0 and the largest to a probability of 1.

Implements algorithm 7 from Hyndman, R. J. and Fan, Y., Sample quantiles in Statistical Packages (R's default quantile function)

Template Parameters

T	An Eigen type with either fixed rows or columns at compile time, or std::vector<double>
Tp	Type of probabilities vector

Parameters

samples_vec	Numeric vector whose sample quantiles are wanted
ps	Vector of probability with value between 0 and 1.

Returns

Sample quantiles, one for each p in ps.

Exceptions

std::invalid_argument	If any of the values are NaN or size 0.
std::domain_error	If p<0 or p>1 for any p in ps.

Definition at line 81 of file quantile.hpp.

template<typename C , require_container_t< C > * = nullptr>

int stan::math::rank ( const C &  v, int  s  )

inline

Return the number of components of v less than v[s].

Template Parameters

C	container type

Parameters

[in]	v	input vector
[in]	s	position in vector

Returns

number of components of v less than v[s].

Exceptions

std::out_of_range

if s is out of range.

Definition at line 20 of file rank.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

int stan::math::rank	(	const T &	v,
		int	s
	)

Return the number of components of v less than v[s].

Template Parameters

C	container type

Parameters

[in]	v	input vector
[in]	s	position in vector

Returns

number of components of v less than v[s].

Exceptions

std::out_of_range

if s is out of range.

Definition at line 24 of file rank.hpp.

template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>

return_type_t<T_y, T_scale> stan::math::rayleigh_cdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 25 of file rayleigh_cdf.hpp.

template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>

return_type_t<T_y, T_scale> stan::math::rayleigh_lccdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 23 of file rayleigh_lccdf.hpp.

template<typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>

return_type_t<T_y, T_scale> stan::math::rayleigh_lcdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 25 of file rayleigh_lcdf.hpp.

template<bool propto, typename T_y , typename T_scale , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_scale > * = nullptr>

return_type_t<T_y, T_scale> stan::math::rayleigh_lpdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 25 of file rayleigh_lpdf.hpp.

template<typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_lpdf ( const T_y &  y, const T_scale &  sigma  )

inline

Definition at line 82 of file rayleigh_lpdf.hpp.

template<typename T , require_var_vector_t< T > * = nullptr>

auto stan::math::read_corr_L	(	const T &	CPCs,
		size_t	K
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

It is generally better to work with the Cholesky factor rather than the correlation matrix itself when the determinant, inverse, etc. of the correlation matrix is needed for some statistical calculation.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 36 of file read_corr_L.hpp.

template<typename T , require_eigen_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L	(	const T &	CPCs,
		size_t	K
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

It is generally better to work with the Cholesky factor rather than the correlation matrix itself when the determinant, inverse, etc. of the correlation matrix is needed for some statistical calculation.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T	type of the array (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 37 of file read_corr_L.hpp.

template<typename T , require_eigen_vector_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L	(	const T &	CPCs,
		size_t	K,
		value_type_t< T > &	log_prob
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

The implementation is Ben Goodrich's Cholesky factor-based approach to the C-vine method of:

• Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, Generating random correlation matrices based on vines and extended onion method Journal of Multivariate Analysis 100 (2009) 1989–2001

Template Parameters

T	type of the array (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Cholesky factor of correlation matrix for specified partial correlations.

Definition at line 101 of file read_corr_L.hpp.

template<typename T1 , typename T2 , require_var_vector_t< T1 > * = nullptr, require_stan_scalar_t< T2 > * = nullptr>

auto stan::math::read_corr_L	(	const T1 &	CPCs,
		size_t	K,
		T2 &	log_prob
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

The implementation is Ben Goodrich's Cholesky factor-based approach to the C-vine method of:

• Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, Generating random correlation matrices based on vines and extended onion method Journal of Multivariate Analysis 100 (2009) 1989–2001

Template Parameters

T	type of input vector (must be a var_value<S> where S inherits from EigenBase)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Cholesky factor of correlation matrix for specified partial correlations.

Definition at line 135 of file read_corr_L.hpp.

template<typename T_CPCs , require_eigen_vector_t< T_CPCs > * = nullptr>

Eigen::Matrix<value_type_t<T_CPCs>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix	(	const T_CPCs &	CPCs,
		size_t	K
	)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T_CPCs

type of the array (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 27 of file read_corr_matrix.hpp.

template<typename T_CPCs , require_var_vector_t< T_CPCs > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::read_corr_matrix ( const T_CPCs &  CPCs, size_t  K  )

inline

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T_CPCs

type of input vector (must be a var_value<T> where T inherits from EigenBase)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 27 of file read_corr_matrix.hpp.

template<typename T_CPCs , require_var_vector_t< T_CPCs > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::read_corr_matrix ( const T_CPCs &  CPCs, size_t  K, scalar_type_t< T_CPCs > &  log_prob  )

inline

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

It is usually preferable to utilize the version that returns the Cholesky factor of the correlation matrix rather than the correlation matrix itself in statistical calculations.

Template Parameters

T_CPCs

type of input vector (must be a var_value<T> where T inherits from EigenBase)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Correlation matrix for specified partial correlations.

Definition at line 56 of file read_corr_matrix.hpp.

template<typename T_CPCs , require_eigen_vector_t< T_CPCs > * = nullptr>

Eigen::Matrix<value_type_t<T_CPCs>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix	(	const T_CPCs &	CPCs,
		size_t	K,
		value_type_t< T_CPCs > &	log_prob
	)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

It is usually preferable to utilize the version that returns the Cholesky factor of the correlation matrix rather than the correlation matrix itself in statistical calculations.

Template Parameters

T_CPCs

type of the array (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Correlation matrix for specified partial correlations.

Definition at line 56 of file read_corr_matrix.hpp.

template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>

Eigen::Matrix<value_type_t<T_CPCs>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_L	(	const T_CPCs &	CPCs,
		const T_sds &	sds,
		value_type_t< T_CPCs > &	log_prob
	)

This is the function that should be called prior to evaluating the density of any elliptical distribution.

Template Parameters

T_CPCs	type of T_CPCs vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)
T_sds	type of sds vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Cholesky factor of covariance matrix for specified partial correlations.

Definition at line 30 of file read_cov_L.hpp.

template<typename T_CPCs , typename T_sds , require_any_var_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>

auto stan::math::read_cov_L ( const T_CPCs &  CPCs, const T_sds &  sds, scalar_type_t< T_CPCs > &  log_prob  )

inline

This is the function that should be called prior to evaluating the density of any elliptical distribution.

Template Parameters

T_CPCs	type of CPCs vector (must be a var_value<T> where T inherits from EigenBase)
T_sds	type of sds vector (must be a var_value<T> where T inherits from EigenBase)

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Cholesky factor of covariance matrix for specified partial correlations.

Definition at line 32 of file read_cov_L.hpp.

template<typename T_CPCs , typename T_sds , require_all_var_vector_t< T_CPCs, T_sds > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::read_cov_matrix	(	const T_CPCs &	CPCs,
		const T_sds &	sds,
		scalar_type_t< T_CPCs > &	log_prob
	)

A generally worse alternative to call prior to evaluating the density of an elliptical distribution.

Template Parameters

T_CPCs	type of CPCs vector (must be a var_value<T> where T inherits from EigenBase)
T_sds	type of sds vector (must be a var_value<T> where T inherits from EigenBase)

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Covariance matrix for specified partial correlations.

Definition at line 29 of file read_cov_matrix.hpp.

template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>

Eigen::Matrix<value_type_t<T_CPCs>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix	(	const T_CPCs &	CPCs,
		const T_sds &	sds,
		value_type_t< T_CPCs > &	log_prob
	)

A generally worse alternative to call prior to evaluating the density of an elliptical distribution.

Template Parameters

T_CPCs	type of T_CPCs vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)
T_sds	type of sds vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Covariance matrix for specified partial correlations.

Definition at line 29 of file read_cov_matrix.hpp.

template<typename T_CPCs , typename T_sds , require_all_var_vector_t< T_CPCs, T_sds > * = nullptr>

var_value<Eigen::MatrixXd> stan::math::read_cov_matrix ( const T_CPCs &  CPCs, const T_sds &  sds  )

inline

Builds a covariance matrix from CPCs and standard deviations.

Template Parameters

T_CPCs	type of CPCs vector (must be a var_value<T> where T inherits from EigenBase)
T_sds	type of sds vector (must be a var_value<T> where T inherits from EigenBase)

Parameters

CPCs	in (-1, 1)
sds	in (0, inf)

Definition at line 46 of file read_cov_matrix.hpp.

template<typename T_CPCs , typename T_sds , require_all_eigen_vector_t< T_CPCs, T_sds > * = nullptr, require_vt_same< T_CPCs, T_sds > * = nullptr>

Eigen::Matrix<value_type_t<T_CPCs>, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix	(	const T_CPCs &	CPCs,
		const T_sds &	sds
	)

Builds a covariance matrix from CPCs and standard deviations.

Template Parameters

T_CPCs	type of T_CPCs vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)
T_sds	type of sds vector (must be derived from Eigen::ArrayBase and have one compile-time dimension equal to 1)

Parameters

CPCs	in (-1, 1)
sds	in (0, inf)

Definition at line 50 of file read_cov_matrix.hpp.

template<typename EigFvar , typename EigOut >

void stan::math::read_fvar ( const EigFvar &  FvarMat, EigOut &  ValMat, EigOut &  DMat  )

inline

Function applying the read_fvar_functor to extract the values and tangets of a given fvar matrix into separate matrices.

Template Parameters

EigFvar	type of the Eigen container of fvar.
EigOut	type of the Eigen containers to copy to

Parameters

[in]	FvarMat	Input Eigen container of fvar.
[in]	ValMat	Output Eigen container of values.
[in]	DMat	Output Eigen container of tangents.

Definition at line 44 of file read_fvar.hpp.

template<typename EigRev , typename EigDbl >

void stan::math::read_val_adj ( const EigRev &  VarMat, EigDbl &  ValMat, EigDbl &  AdjMat  )

inline

Function applying the val_adj_functor to extract the values and adjoints of a given var or vari matrix into separate matrices.

Template Parameters

EigRev	type of the Eigen container of var or vari.
EigDbl	type of the Eigen container of doubles to be copied to.

Parameters

[in]	VarMat	Input Eigen container of var.
[in]	ValMat	Output Eigen container of values.
[in]	AdjMat	Output Eigen container of adjoints.

Definition at line 158 of file read_var.hpp.

template<typename EigVar , typename EigVari , typename EigDbl >

void stan::math::read_vi_adj ( const EigVar &  VarMat, EigVari &  VariMat, EigDbl &  AdjMat  )

inline

Function applying the vi_adj_functor to extract the varis and and adjoints of a given var matrix into separate matrices.

Template Parameters

EigVar	type of the Eigen container of var.
EigDbl	type of the Eigen container of doubles to be copied to.

Parameters

[in]	VarMat	Input Eigen container of var.
[in]	VariMat	Output Eigen container of vari.
[in]	AdjMat	Output Eigen container of adjoints.

Definition at line 193 of file read_var.hpp.

template<typename EigVar , typename EigVari , typename EigDbl >

void stan::math::read_vi_val ( const EigVar &  VarMat, EigVari &  VariMat, EigDbl &  ValMat  )

inline

Function applying the vi_val_functor to extract the varis and and values of a given var matrix into separate matrices.

Template Parameters

EigVar	type of the Eigen container of var.
EigDbl	type of the Eigen container of doubles to be copied to.

Parameters

[in]	VarMat	Input Eigen container of var.
[in]	VariMat	Output Eigen container of vari.
[in]	ValMat	Output Eigen container of values.

Definition at line 175 of file read_var.hpp.

template<typename EigVar , typename EigVari , typename EigDbl >

void stan::math::read_vi_val_adj ( const EigVar &  VarMat, EigVari &  VariMat, EigDbl &  ValMat, EigDbl &  AdjMat  )

inline

Function applying the vi_val_adj_functor to extract the vari*, values, and adjoints of a given var matrix into separate matrices.

Template Parameters

EigVar	type of the Eigen container of var.
EigVari	type of the Eigen container of vari to be copied to.
EigDbl	type of the Eigen container of doubles to be copied to.

Parameters

[in]	VarMat	Input Eigen container of var.
[in]	VariMat	Output Eigen container of vari.
[in]	ValMat	Output Eigen container of values.
[in]	AdjMat	Output Eigen container of tangents.

Definition at line 139 of file read_var.hpp.

template<typename T , require_autodiff_t< T > >

T stan::math::real

(

const std::complex< T > &

z

)

Return the real component of the complex argument.

Template Parameters

T	value type of complex argument

Parameters

[in]

z

complex value whose real component is extracted

Returns

real component of argument

Definition at line 18 of file real.hpp.

static void stan::math::recover_memory ( )

inlinestatic

Recover memory used for all variables for reuse.

Exceptions

std::logic_error

if empty_nested() returns false

Definition at line 18 of file recover_memory.hpp.

static void stan::math::recover_memory_nested ( )

inlinestatic

Recover only the memory used for the top nested call.

If there is nothing on the nested stack, then a std::logic_error exception is thrown.

It is preferred to use the nested_rev_autodiff class for nested autodiff as it handles recovery of memory automatically.

Exceptions

std::logic_error

if empty_nested() returns true

Definition at line 23 of file recover_memory_nested.hpp.

template<typename ReduceFunction , typename Vec , typename = require_vector_like_t<Vec>, typename... Args>

auto stan::math::reduce_sum ( Vec &&  vmapped, int  grainsize, std::ostream *  msgs, Args &&...  args  )

inline

Call an instance of the function ReduceFunction on every element of an input sequence and sum these terms.

This defers to reduce_sum_impl for the appropriate implementation

ReduceFunction must define an operator() with the same signature as: T f(Vec&& vmapped_subset, int start, int end, std::ostream* msgs, Args&&... args)

ReduceFunction must be default constructible without any arguments

grainsize must be greater than or equal to 1

Template Parameters

ReduceFunction	Type of reducer function
ReturnType	An arithmetic type
Vec	Type of sliced argument
Args	Types of shared arguments

Parameters

	vmapped	Vector containing one element per term of sum
	grainsize	Suggested grainsize for tbb
[in,out]	msgs	The print stream for warning messages
	args	Shared arguments used in every sum term

Returns

Sum of terms

Definition at line 199 of file reduce_sum.hpp.

template<typename ReduceFunction , typename Vec , typename = require_vector_like_t<Vec>, typename... Args>

auto stan::math::reduce_sum_static	(	Vec &&	vmapped,
		int	grainsize,
		std::ostream *	msgs,
		Args &&...	args
	)

Call an instance of the function ReduceFunction on every element of an input sequence and sum these terms.

This defers to reduce_sum_impl for the appropriate implementation

ReduceFunction must define an operator() with the same signature as: T f(Vec&& vmapped_subset, int start, int end, std::ostream* msgs, Args&&... args)

ReduceFunction must be default constructible without any arguments

grainsize must be greater than or equal to 1

If STAN_THREADS is not defined, do all the work with one ReduceFunction call.

Template Parameters

ReduceFunction	Type of reducer function
ReturnType	An arithmetic type
Vec	Type of sliced argument
Args	Types of shared arguments

Parameters

	vmapped	Sliced arguments used only in some sum terms
	grainsize	Suggested grainsize for tbb
[in,out]	msgs	The print stream for warning messages
	args	Shared arguments used in every sum term

Returns

Sum of terms

Definition at line 45 of file reduce_sum_static.hpp.

template<typename T_ret , typename In , require_std_vector_t< T_ret > * = nullptr>

std::vector<plain_type_t<In> > stan::math::rep_array ( const In &  x, int  n  )

inline

Definition at line 12 of file rep_array.hpp.

template<typename In >

std::vector<plain_type_t<In> > stan::math::rep_array ( const In &  x, int  n  )

inline

Definition at line 18 of file rep_array.hpp.

template<typename In >

std::vector<std::vector<plain_type_t<In> > > stan::math::rep_array ( const In &  x, int  m, int  n  )

inline

Definition at line 23 of file rep_array.hpp.

template<typename In >

std::vector<std::vector<std::vector<plain_type_t<In> > > > stan::math::rep_array ( const In &  x, int  k, int  m, int  n  )

inline

Definition at line 33 of file rep_array.hpp.

template<typename Ret , typename T , require_eigen_matrix_dynamic_vt< is_stan_scalar, Ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_matrix ( const T &  x, int  m, int  n  )

inline

Implementation of rep_matrix returning an Eigen matrix with scalar type equal to the input scalar type.

Impl of rep_matrix returning an var_value<Eigen::Matrix> with a var scalar type.

Template Parameters

Ret	An Eigen type.
T	A Scalar type.

Parameters

x	A Scalar whose values are propogated to all values in the return matrix.
m	Number or rows.
n	Number of columns.

Template Parameters

Ret	A var_value with inner Eigen type.
T	A Scalar type.

Parameters

x	A Scalar whose values are propogated to all values in the return matrix.
m	Number or rows.
n	Number of columns.

Definition at line 24 of file rep_matrix.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_matrix ( const T &  x, int  m, int  n  )

inline

Default Implementation of rep_matrix returning an Eigen matrix with scalar type equal to the input scalar type.

Template Parameters

T	A Scalar type.

Parameters

x	A Scalar whose values are propogated to all values in the return matrix.
m	Number or rows.
n	Number of columns.

Definition at line 40 of file rep_matrix.hpp.

template<typename Vec , require_eigen_vector_t< Vec > * = nullptr>

auto stan::math::rep_matrix ( const Vec &  x, int  n  )

inline

Implementation of rep_matrix returning an Eigen matrix from an Eigen vector.

Impl of rep_matrix returning a var_value<Eigen::Matrix> from a var_value with an inner Eigen vector type.

Template Parameters

Vec	An Eigen vector.

Parameters

x	An Eigen vector. For Row vectors the values are replacated rowwise. and for column vectors the values are repliacated colwise.
n	Number of rows or columns.

Template Parameters

Ret	A var_value with inner Eigen dynamic matrix type.
Vec	A var_value with an inner Eigen vector type.

Parameters

x	A var_value with inner Eigen vector type. For Row vectors the values are replacated rowwise and for column vectors the values are repliacated colwise.
n	Number of rows or columns.

Definition at line 54 of file rep_matrix.hpp.

template<typename T_ret , typename T , require_eigen_row_vector_t< T_ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_row_vector ( const T &  x, int  n  )

inline

Definition at line 13 of file rep_row_vector.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_row_vector ( const T &  x, int  n  )

inline

Definition at line 18 of file rep_row_vector.hpp.

template<typename T_ret , require_var_matrix_t< T_ret > * = nullptr, require_eigen_row_vector_t< value_type_t< T_ret >> * = nullptr>

auto stan::math::rep_row_vector ( var  x, int  n  )

inline

Overload for var_value<Vector>.

Template Parameters

T_ret	The user supplied return type.
T	A double or var type

Parameters

x	The type to be propogated through the new vector.
n	The size of the new vector.

Definition at line 22 of file rep_row_vector.hpp.

template<typename T_ret , typename T , require_eigen_col_vector_t< T_ret > * = nullptr, require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_vector ( const T &  x, int  n  )

inline

Definition at line 13 of file rep_vector.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

auto stan::math::rep_vector ( const T &  x, int  n  )

inline

Definition at line 18 of file rep_vector.hpp.

template<typename T_ret , require_var_matrix_t< T_ret > * = nullptr, require_eigen_col_vector_t< value_type_t< T_ret >> * = nullptr>

auto stan::math::rep_vector ( var  x, int  n  )

inline

Overload for var_value<Vector>.

Template Parameters

T_ret	The user supplied return type.
T	A double or var type

Parameters

x	The type to be propogated through the new vector.
n	The size of the new vector.

Definition at line 22 of file rep_vector.hpp.

template<typename T >

void stan::math::resize ( T &  x, std::vector< int >  dims  )

inline

Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices.

Template Parameters

T	type of object being resized

Parameters

x	Array-like object to resize.
dims	New dimensions.

Definition at line 43 of file resize.hpp.

template<typename T >

std::vector<T> stan::math::reverse ( const std::vector< T > &  x )

inline

Return a copy of the specified array in reversed order.

Template Parameters

T	type of elements in the array

Parameters

x	array to reverse

Returns

Array in reversed order.

Definition at line 20 of file reverse.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::reverse ( T_x &&  x )

inline

Return reversed view into the specified vector or row vector.

Template Parameters

T	type of expression

Parameters

x	vector or row vector to reverse

Returns

reversed vector or row vector

Definition at line 20 of file reverse.hpp.

template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_any_var_t< T > * = nullptr>

auto stan::math::reverse ( const T &  m )

inline

Return reversed view into the specified vector or row vector.

Template Parameters

T	type of expression

Parameters

m	vector or row vector to reverse

Returns

reversed vector or row vector

Definition at line 23 of file reverse.hpp.

template<typename T , typename = require_vector_t<T>>

auto stan::math::reverse ( const T &  x )

inline

Return a copy of the specified vector or row vector in reversed order.

Template Parameters

T	type of container (vector or row vector)

Parameters

x	vector or row vector to reverse

Returns

Vector or row vector in reversed order.

Definition at line 35 of file reverse.hpp.

template<typename F >

void stan::math::reverse_pass_callback ( F &&  functor )

inline

Puts a callback on the autodiff stack to be called in reverse pass.

The intended use case is for the callable to ba a lambda function that captures any arguments it needs to work with. All captured values must be trivially destructible or they will leak memory! to_AD_stack() function can be used to ensure that.

Template Parameters

F	type of callable

Parameters

functor

funtor or other callable to call in the reverse pass

Definition at line 38 of file reverse_pass_callback.hpp.

var stan::math::rising_factorial ( const var &  a, int  b  )

inline

Definition at line 25 of file rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::rising_factorial ( const fvar< T > &  x, int  n  )

inline

Return autodiff variable with the gradient and result of the rising factorial function applied to the inputs.

Will throw for NaN x and for negative n, as implemented in primitive function.

Template Parameters

T	inner type of the fvar

Parameters

x	Argument.
n	Argument

Returns

tangent of rising factorial at arguments.

Definition at line 26 of file rising_factorial.hpp.

template<typename T , require_arithmetic_t< T > * = nullptr>

return_type_t<T> stan::math::rising_factorial ( const T &  x, int  n  )

inline

Return the rising factorial function evaluated at the inputs.

Template Parameters

T	type of the first argument

Parameters

x	first argument
n	second argument

Returns

Result of rising factorial function.

Exceptions

std::domain_error	if x is NaN
std::domain_error	if n is negative

\[ \mbox{rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)} \]

\[ \frac{\partial \, x^{(n)}}{\partial x} = x^{(n)}(\Psi(x+n)-\Psi(x)) \]

\[ \frac{\partial \, x^{(n)}}{\partial n} = (x)_n\Psi(x+n) \]

Definition at line 63 of file rising_factorial.hpp.

template<typename T1 , typename T2 , require_any_container_t< T1, T2 > * = nullptr>

auto stan::math::rising_factorial ( const T1 &  a, const T2 &  b  )

inline

Enables the vectorised application of the rising_factorial function, when the first and/or second arguments are containers.

Template Parameters

T1	type of first input
T2	type of second input

Parameters

a	First input
b	Second input

Returns

rising_factorial function applied to the two inputs.

Definition at line 81 of file rising_factorial.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::round ( const var_value< T > &  A )

inline

Returns the elementwise round() of the input var_value<matrix_cl<double>>.

Parameters

A	input var_value<matrix_cl<double>>

Returns

Elementwise round() of the input argument.

Definition at line 22 of file round.hpp.

template<typename T >

fvar<T> stan::math::round ( const fvar< T > &  x )

inline

Return the closest integer to the specified argument, with halfway cases rounded away from zero.

The derivative is always zero.

Template Parameters

T	inner type of the fvar

Parameters

x

Argument.

Returns

The rounded value of the argument.

Definition at line 24 of file round.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::round ( const Container &  x )

inline

Vectorized version of round().

Template Parameters

Container

type of container

Parameters

x

container

Returns

Rounded value of each value in x.

Definition at line 39 of file round.hpp.

var stan::math::round ( const var &  a )

inline

Returns the rounded form of the specified variable (C99).

The derivative is zero everywhere but numbers half way between whole numbers, so for convenience the derivative is defined to be everywhere zero,

\(\frac{d}{dx} \mbox{round}(x) = 0\).

\[ \mbox{round}(x) = \begin{cases} \lceil x \rceil & \mbox{if } x-\lfloor x\rfloor \geq 0.5 \\ \lfloor x \rfloor & \mbox{if } x-\lfloor x\rfloor < 0.5 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{round}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Rounded variable.

Definition at line 43 of file round.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::round ( const Container &  x )

inline

Version of round() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Rounded value of each value in x.

Definition at line 53 of file round.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::row ( const T &  m, size_t  i  )

inline

Return the specified row of the specified matrix, using start-at-1 indexing.

This is equivalent to calling m.row(i - 1) and assigning the resulting template expression to a row vector.

Template Parameters

T	type of the matrix

Parameters

m	Matrix.
i	Row index (count from 1).

Returns

Specified row of the matrix.

Exceptions

std::out_of_range

if i is out of range.

Definition at line 24 of file row.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

int stan::math::rows ( const T &  m )

inline

Return the number of rows in the specified matrix, vector, or row vector.

Template Parameters

T	type of the matrix

Parameters

[in]

m

Input matrix, vector, or row vector.

Returns

Number of rows.

Definition at line 19 of file rows.hpp.

template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::rows_dot_product ( T_a &&  a, T_b &&  b  )

inline

Returns the dot product of rows of the specified matrices.

Template Parameters

T_a	type of the first matrix
T_b	type of the second matrix

Parameters

a	Matrix of first vectors.
b	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the matrices are not the same size

Definition at line 27 of file rows_dot_product.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>

Eigen::Matrix<return_type_t<Mat1, Mat2>, Mat1::RowsAtCompileTime, 1> stan::math::rows_dot_product ( const Mat1 &  v1, const Mat2 &  v2  )

inline

Returns the dot product of rows of the specified matrices.

Template Parameters

Mat1	type of the first matrix (must be derived from Eigen::MatrixBase)
Mat2	type of the second matrix (must be derived from Eigen::MatrixBase)

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the matrices are not the same size

Definition at line 28 of file rows_dot_product.hpp.

template<typename T1 , typename T2 , require_any_var_t< T1, T2 > * = nullptr, require_all_nonscalar_prim_or_rev_kernel_expression_t< T1, T2 > * = nullptr>

var_value<matrix_cl<double> > stan::math::rows_dot_product ( T1 &&  v1, T2 &&  v2  )

inline

Returns the dot product of rows of the specified matrices.

Template Parameters

T1	type of the first matrix
T2	type of the second matrix

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::invalid_argument

If the vectors are not the same size

Definition at line 30 of file rows_dot_product.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_any_eigen_vt< is_var, Mat1, Mat2 > * = nullptr>

Eigen::Matrix<var, Mat1::RowsAtCompileTime, 1> stan::math::rows_dot_product ( const Mat1 &  v1, const Mat2 &  v2  )

inline

Returns the dot product of rows of the specified matrices.

Template Parameters

Mat1	type of the first matrix (must be derived from Eigen::MatrixBase)
Mat2	type of the second matrix (must be derived from Eigen::MatrixBase)

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 33 of file rows_dot_product.hpp.

template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>

auto stan::math::rows_dot_product ( const Mat1 &  v1, const Mat2 &  v2  )

inline

Returns the dot product of rows of the specified matrices.

This overload is used when at least one of Mat1 and Mat2 is a var_value<T> where T inherits from EigenBase. The other argument can be another var_value or a type that inherits from EigenBase.

Template Parameters

Mat1	type of the first matrix
Mat2	type of the second matrix

Parameters

v1	Matrix of first vectors.
v2	Matrix of second vectors.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 63 of file rows_dot_product.hpp.

template<typename Mat , require_eigen_vt< is_var, Mat > * = nullptr>

Eigen::Matrix<var, Mat::RowsAtCompileTime, 1> stan::math::rows_dot_self ( const Mat &  x )

inline

Returns the dot product of each row of a matrix with itself.

Template Parameters

Mat	An Eigen matrix with a var scalar type.

Parameters

x

Matrix.

Definition at line 20 of file rows_dot_self.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_st_var< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, T::RowsAtCompileTime, 1> stan::math::rows_dot_self ( const T &  x )

inline

Returns the dot product of each row of a matrix with itself.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

x

matrix

Definition at line 20 of file rows_dot_self.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::rows_dot_self ( const var_value< T > &  v )

inline

Returns the dot product of each row of a matrix with itself.

Template Parameters

T	type of the matrix

Parameters

v

Matrix.

Definition at line 22 of file rows_dot_self.hpp.

template<typename T_a , require_all_kernel_expressions_and_none_scalar_t< T_a > * = nullptr>

auto stan::math::rows_dot_self ( T_a &&  a )

inline

Returns the dot product of each row of a matrix with itself.

Template Parameters

T_a	type of the matrix

Parameters

a

matrix

Returns

Dot product of the vectors.

Definition at line 23 of file rows_dot_self.hpp.

template<typename Mat , require_var_matrix_t< Mat > * = nullptr>

auto stan::math::rows_dot_self ( const Mat &  x )

inline

Returns the dot product of row row of a matrix with itself.

Template Parameters

Mat	A var_value<> with an inner matrix type.

Parameters

x

Matrix.

Definition at line 36 of file rows_dot_self.hpp.

template<typename T , typename = require_all_kernel_expressions_and_none_scalar_t<T>>

auto stan::math::rowwise_optional_broadcast ( T &&  a )

inline

Broadcast an expression in rowwise dimmension if the number of columns equals to 1.

In that case further expressions can use this expression as if had any number of columns, repeating the values.

Broadcasting evaluates argument expression multiple times. For performance reasons don't broadcast slow operations. Instead evaluate them in a separate kernel.

Template Parameters

T	type of input expression

Parameters

a	input expression

Returns

broadcast expression

Definition at line 222 of file optional_broadcast.hpp.

template<typename... Pargs>

vari ** stan::math::save_varis ( vari **  dest, const var &  x, Pargs &&...  args  )

inline

Save the vari pointer in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer.

Template Parameters

Pargs

Types of remaining arguments

Parameters

[in,out]	dest	Pointer to where vari pointers are saved
[in]	x	A var
[in]	args	Additional arguments to have their varis saved

Returns

Final position of dest pointer

Definition at line 51 of file save_varis.hpp.

template<typename VarVec , require_std_vector_vt< is_var, VarVec > * = nullptr, typename... Pargs>

vari ** stan::math::save_varis ( vari **  dest, VarVec &&  x, Pargs &&...  args  )

inline

Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer.

Template Parameters

VarVec	A variant of std::vector<var>
Pargs	Types of remaining arguments

Parameters

[in,out]	dest	Pointer to where vari pointers are saved
[in]	x	A std::vector of vars
[in]	args	Additional arguments to have their varis saved

Returns

Final position of dest pointer

Definition at line 71 of file save_varis.hpp.

template<typename VecContainer , require_std_vector_st< is_var, VecContainer > * = nullptr, require_std_vector_vt< is_container, VecContainer > * = nullptr, typename... Pargs>

vari ** stan::math::save_varis ( vari **  dest, VecContainer &&  x, Pargs &&...  args  )

inline

Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer.

Template Parameters

VecContainer	std::vector<T> where T is another type containing vars
Pargs	Types of remaining arguments

Parameters

[in,out]	dest	Pointer to where vari pointers are saved
[in]	x	A std::vector of containers containing of vars
[in]	args	Additional arguments to have their varis saved

Returns

Final position of dest pointer

Definition at line 93 of file save_varis.hpp.

template<typename EigT , require_eigen_vt< is_var, EigT > * = nullptr, typename... Pargs>

vari ** stan::math::save_varis ( vari **  dest, EigT &&  x, Pargs &&...  args  )

inline

Save the vari pointers in x into the memory pointed to by dest, increment the dest storage pointer, recursively call save_varis on the rest of the arguments, and return the final value of the dest storage pointer.

Template Parameters

EigT	An Eigen type with var value type
Pargs	Types of remaining arguments

Parameters

[in,out]	dest	Pointer to where vari pointers are saved
[in]	x	An Eigen container of vars
[in]	args	Additional arguments to have their varis saved

Returns

Final position of dest pointer

Definition at line 114 of file save_varis.hpp.

template<typename Arith , require_st_arithmetic< Arith > * = nullptr, typename... Pargs>

vari ** stan::math::save_varis ( vari **  dest, Arith &&  x, Pargs &&...  args  )

inline

Ignore arithmetic types.

Recursively call save_varis on the rest of the arguments and return the final value of the dest storage pointer.

Template Parameters

Arith	An arithmetic type
Pargs	Types of remaining arguments

Parameters

[in,out]	dest	Pointer to where vari pointers are saved
[in]	x	An argument not containing vars
[in]	args	Additional arguments to have their varis saved

Returns

Final position of dest pointer

Definition at line 135 of file save_varis.hpp.

vari ** stan::math::save_varis ( vari **  dest )

inline

End save_varis recursion and return pointer.

Parameters

dest

Pointer

Definition at line 144 of file save_varis.hpp.

template<typename T , typename = require_autodiff_t<T>>

double stan::math::scalbn	(	const T &	x,
		int	n
	)

Definition at line 12 of file scalbn.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< std::is_arithmetic, EigMat1, EigMat2 > * = nullptr>

Eigen::Matrix<double, Eigen::Dynamic, EigMat2::ColsAtCompileTime> stan::math::scale_matrix_exp_multiply ( const double &  t, const EigMat1 &  A, const EigMat2 &  B  )

inline

Return product of exp(At) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double.

Specialized for double values for efficiency.

Template Parameters

EigMat1	type of the first matrix
EigMat2	type of the second matrix

Parameters

[in]	A	Matrix
[in]	B	Matrix
[in]	t	double

Returns

exponential of At multiplied by B

Definition at line 30 of file scale_matrix_exp_multiply.hpp.

template<typename Tt , typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_autodiff_t< Tt, value_type_t< EigMat1 >, value_type_t< EigMat2 >> * = nullptr>

Eigen::Matrix<return_type_t<Tt, EigMat1, EigMat2>, Eigen::Dynamic, EigMat2::ColsAtCompileTime> stan::math::scale_matrix_exp_multiply ( const Tt &  t, const EigMat1 &  A, const EigMat2 &  B  )

inline

Return product of exp(At) and B, where A is a NxN matrix, B is a NxCb matrix and t is a scalar.

Generic implementation when arguments are not all double.

Template Parameters

Tt	type of t
EigMat1	type of the first matrix
EigMat2	type of the second matrix

Parameters

[in]	A	Matrix
[in]	B	Matrix
[in]	t	double

Returns

exponential of At multiplied by B

Definition at line 60 of file scale_matrix_exp_multiply.hpp.

void stan::math::scaled_add ( std::vector< double > &  x, const std::vector< double > &  y, double  lambda  )

inline

Definition at line 11 of file scaled_add.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lccdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_scale &	s
	)

Definition at line 25 of file scaled_inv_chi_square_lccdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lcdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_scale &	s
	)

Definition at line 25 of file scaled_inv_chi_square_lcdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lpdf ( const T_y &  y, const T_dof &  nu, const T_scale &  s  )

inline

Definition at line 176 of file scaled_inv_chi_square_lpdf.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::sd ( const var_value< T > &  A )

inline

Return the sample standard deviation of the var_value matrix.

Template Parameters

T	Input type

Parameters

[in]

A

input matrix

Returns

sample standard deviation of specified matrix

Exceptions

domain

error size is not greater than zero.

Definition at line 24 of file sd.hpp.

template<typename T , require_container_t< T > * = nullptr, require_not_st_var< T > * = nullptr>

auto stan::math::sd ( const T &  m )

inline

Returns the unbiased sample standard deviation of the coefficients in the specified std vector, column vector, row vector, or matrix.

Template Parameters

T	type of the container

Parameters

m	Specified container.

Returns

Sample variance.

Definition at line 26 of file sd.hpp.

template<typename T , require_eigen_st< is_var, T > * = nullptr>

var stan::math::sd

(

const T &

x

)

Return the sample standard deviation of a variable which inherits from EigenBase.

Return the sample standard deviation of the var_value matrix.

Template Parameters

T	Input type

Parameters

[in]

x

input

Returns

sample standard deviation

Exceptions

domain

error size is not greater than zero.

Template Parameters

T	Input type

Parameters

[in]

x

input matrix

Returns

sample standard deviation of specified matrix

Exceptions

domain

error size is not greater than zero.

Definition at line 26 of file sd.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

double stan::math::sd ( const T &  a )

inline

Returns the unbiased sample standard deviation of the coefficients in the specified std vector, column vector, row vector, or matrix.

Template Parameters

T	type of the container

Parameters

a	Specified container.

Returns

Sample standard deviation.

Definition at line 26 of file sd.hpp.

template<typename T , require_std_vector_st< is_var, T > * = nullptr>

auto stan::math::sd

(

const T &

m

)

Return the sample standard deviation of the specified std vector, column vector, row vector, matrix, or std vector of any of these types.

Template Parameters

T	Input type

Parameters

[in]

m

input matrix

Returns

sample standard deviation

Exceptions

domain

error size is not greater than zero.

Definition at line 95 of file sd.hpp.

template<typename Vec , require_vector_t< Vec > * = nullptr>

auto stan::math::segment ( const Vec &  v, size_t  i, size_t  n  )

inline

Return the specified number of elements as a row/column vector starting from the specified element - 1 of the specified row/column vector.

Template Parameters

T	type of the vector

Definition at line 18 of file segment.hpp.

template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>

auto stan::math::segment ( T_x &&  x, size_t  i, size_t  n  )

inline

Return the specified number of elements as a row/column vector starting from the specified element - 1 of the specified row/column vector.

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x	input kernel generator expression.
i	Starting row/column + 1.
n	Number of rows/columns in segment.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 24 of file segment.hpp.

template<typename T >

std::vector<T> stan::math::segment	(	const std::vector< T > &	sv,
		size_t	i,
		size_t	n
	)

Definition at line 30 of file segment.hpp.

static void stan::math::set_zero_all_adjoints ( )

static

Reset all adjoint values in the stack to zero.

Definition at line 14 of file set_zero_all_adjoints.hpp.

static void stan::math::set_zero_all_adjoints_nested ( )

static

Reset all adjoint values in the top nested portion of the stack to zero.

It is preferred to use the nested_rev_autodiff class for nested autodiff class as it handles recovery of memory automatically.

Definition at line 20 of file set_zero_all_adjoints_nested.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

int stan::math::sign ( const T &  z )

inline

Definition at line 12 of file sign.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

auto stan::math::sign ( const T &  x )

inline

Returns signs of the arguments.

Return the elementwise application of sign() to specified argument container.

Template Parameters

T	type of the argument (matrix_cl or kernel generator expression)

Parameters

x	the argument

Returns

sign of x

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise sign of members of container.

Definition at line 18 of file sign.hpp.

template<typename ADType , require_autodiff_t< ADType > * = nullptr>

bool stan::math::signbit ( ADType &&  v )

inline

Return true if the specified argument is negative and false otherwise.

Overloads std::signbit from <cmath> for argument-dependent lookup.

Template Parameters

ADType

type of argument

Parameters

[in]

v

argument

Returns

true if the argument is negative

Definition at line 21 of file signbit.hpp.

template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr, require_not_st_var< Vec > * = nullptr>

auto stan::math::simplex_constrain ( const Vec &  y )

inline

Return the simplex corresponding to the specified free vector.

A simplex is a vector containing values greater than or equal to 0 that sum to 1. A vector with (K-1) unconstrained values will produce a simplex of size K.

The transform is based on a centered stick-breaking process.

Template Parameters

Vec	type of the vector

Parameters

y	Free vector input of dimensionality K - 1.

Returns

Simplex of dimensionality K.

Definition at line 29 of file simplex_constrain.hpp.

template<typename T , require_rev_col_vector_t< T > * = nullptr>

auto stan::math::simplex_constrain ( const T &  y )

inline

Return the simplex corresponding to the specified free vector.

A simplex is a vector containing values greater than or equal to 0 that sum to 1. A vector with (K-1) unconstrained values will produce a simplex of size K.

The transform is based on a centered stick-breaking process.

Template Parameters

T	Type of vector to constrain

Parameters

y	Free vector input of dimensionality K - 1

Returns

Simplex of dimensionality K

Definition at line 31 of file simplex_constrain.hpp.

template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr, require_not_st_var< Vec > * = nullptr>

auto stan::math::simplex_constrain ( const Vec &  y, value_type_t< Vec > &  lp  )

inline

Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The simplex transform is defined through a centered stick-breaking process.

Template Parameters

Vec	type of the vector

Parameters

y	Free vector input of dimensionality K - 1.
lp	Log probability reference to increment.

Returns

Simplex of dimensionality K.

Definition at line 61 of file simplex_constrain.hpp.

template<typename T , require_rev_col_vector_t< T > * = nullptr>

auto stan::math::simplex_constrain	(	const T &	y,
		scalar_type_t< T > &	lp
	)

Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The simplex transform is defined through a centered stick-breaking process.

Template Parameters

T	type of the vector to constrain

Parameters

y	Free vector input of dimensionality N.
lp	Log probability reference to increment.

Returns

Simplex of dimensionality N + 1.

Definition at line 85 of file simplex_constrain.hpp.

template<bool Jacobian, typename Vec , require_not_std_vector_t< Vec > * = nullptr>

auto stan::math::simplex_constrain	(	const Vec &	y,
		return_type_t< Vec > &	lp
	)

Return the simplex corresponding to the specified free vector.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
Vec	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

[in]	y	free vector
[in,out]	lp	log density accumulator

Returns

simplex of dimensionality one greater than y

Definition at line 101 of file simplex_constrain.hpp.

template<bool Jacobian, typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::simplex_constrain ( const T &  y, return_type_t< T > &  lp  )

inline

Return the simplex corresponding to the specified free vector.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
Vec	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

[in]	y	free vector
[in,out]	lp	log density accumulator

Returns

simplex of dimensionality one greater than y

Definition at line 126 of file simplex_constrain.hpp.

template<typename Vec , require_eigen_col_vector_t< Vec > * = nullptr>

auto stan::math::simplex_free

(

const Vec &

x

)

Return an unconstrained vector that when transformed produces the specified simplex.

It applies to a simplex of dimensionality K and produces an unconstrained vector of dimensionality (K-1).

The simplex transform is defined through a centered stick-breaking process.

Template Parameters

ColVec

type of the simplex (must be a column vector)

Parameters

x	Simplex of dimensionality K.

Returns

Free vector of dimensionality (K-1) that transforms to the simplex.

Exceptions

std::domain_error

if x is not a valid simplex

Definition at line 30 of file simplex_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::simplex_free

(

const T &

x

)

Overload of simplex_free() to untransform each Eigen vector in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase with compile time rows or columns equal to 1.

Parameters

x	The standard vector to untransform.

Definition at line 56 of file simplex_free.hpp.

template<typename T >

fvar<T> stan::math::sin ( const fvar< T > &  x )

inline

Definition at line 14 of file sin.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::sin ( const var_value< T > &  A )

inline

Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise cos() of the input, in radians.

Definition at line 21 of file sin.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::sin ( const std::complex< fvar< T >> &  z )

inline

Return the sine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

sine of the argument

Definition at line 28 of file sin.hpp.

template<typename T , require_not_container_st< std::is_arithmetic, T > * = nullptr, require_not_var_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::sin ( const T &  x )

inline

Vectorized version of sin().

Template Parameters

Container

type of container

Parameters

x	angles in radians

Returns

Sine of each value in x.

Definition at line 42 of file sin.hpp.

var stan::math::sin ( const var &  a )

inline

Return the sine of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sin x = \cos x\).

\[ \mbox{sin}(x) = \begin{cases} \sin(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sin}(x)}{\partial x} = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Sine of variable.

Definition at line 46 of file sin.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::sin ( const Container &  x )

inline

Version of sin() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Sine of each value in x.

Definition at line 56 of file sin.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::sin ( const VarMat &  a )

inline

Return the sine of a radian-scaled variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Variable for radians of angle.

Returns

Sine of variable.

Definition at line 60 of file sin.hpp.

std::complex<var> stan::math::sin ( const std::complex< var > &  z )

inline

Return the sine of the complex argument.

Parameters

[in]

z

argument

Returns

sine of the argument

Definition at line 73 of file sin.hpp.

template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>

auto stan::math::singular_values ( const EigMat &  m )

inline

Return the singular values of the specified matrix.

Adjoint update equation comes from Equation (4) in Differentiable Programming Tensor Networks(H. Liao, J. Liu, et al., arXiv:1903.09650).

Template Parameters

EigMat

type of input matrix

Parameters

m	MxN input matrix

Returns

Singular values of matrix

Definition at line 23 of file singular_values.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, 1> stan::math::singular_values

(

const EigMat &

m

)

Return the vector of the singular values of the specified matrix in decreasing order of magnitude.

See the documentation for svd() for information on the singular values.

Template Parameters

EigMat

type of the matrix

Parameters

m	Specified matrix.

Returns

Singular values of the matrix.

Definition at line 23 of file singular_values.hpp.

template<typename T >

fvar<T> stan::math::sinh ( const fvar< T > &  x )

inline

Definition at line 13 of file sinh.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::sinh ( const var_value< T > &  A )

inline

Returns the elementwise cos() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise cos() of the input, in radians.

Definition at line 21 of file sinh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::sinh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic sine of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic sine of the argument

Definition at line 27 of file sinh.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::sinh ( const Container &  x )

inline

Vectorized version of sinh().

Template Parameters

Container

type of container

Parameters

x

container

Returns

Hyperbolic sine of each variable in x.

Definition at line 41 of file sinh.hpp.

var stan::math::sinh ( const var &  a )

inline

Return the hyperbolic sine of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sinh x = \cosh x\).

\[ \mbox{sinh}(x) = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sinh}(x)}{\partial x} = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic sine of variable.

Definition at line 44 of file sinh.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::sinh ( const Container &  x )

inline

Version of sinh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Hyperbolic sine of each variable in x.

Definition at line 55 of file sinh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::sinh ( const VarMat &  a )

inline

Return the hyperbolic of a radian-scaled variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Variable for radians of angle.

Returns

Hyperbolid Sine of variable.

Definition at line 59 of file sinh.hpp.

std::complex<var> stan::math::sinh ( const std::complex< var > &  z )

inline

Return the hyperbolic sine of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic sine of the argument

Definition at line 72 of file sinh.hpp.

template<typename MatrixT , require_matrix_t< MatrixT > * = nullptr>

size_t stan::math::size_mvt

(

const MatrixT &

)

Definition at line 29 of file size_mvt.hpp.

template<typename MatrixT , require_matrix_t< MatrixT > * = nullptr>

size_t stan::math::size_mvt

(

const std::vector< MatrixT > &

x

)

Definition at line 34 of file size_mvt.hpp.

template<typename T >

bool stan::math::size_zero ( const T &  x )

inline

Returns 1 if input is of length 0, returns 0 otherwise.

Parameters

x

argument

Returns

0 or 1

Definition at line 18 of file size_zero.hpp.

template<typename T , typename... Ts>

bool stan::math::size_zero ( const T &  x, const Ts &...  xs  )

inline

Returns 1 if any inputs are of length 0, returns 0 otherwise.

Parameters

x	first argument
xs	parameter pack of remaining arguments to forward to function

Returns

0 or 1

Definition at line 31 of file size_zero.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_skewness >

return_type_t<T_y, T_loc, T_scale, T_skewness> stan::math::skew_double_exponential_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_skewness &	tau
	)

Definition at line 131 of file skew_double_exponential_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 28 of file skew_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 26 of file skew_normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 28 of file skew_normal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_loc, T_scale, T_shape > * = nullptr>

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 29 of file skew_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma, const T_shape &  alpha  )

inline

Definition at line 119 of file skew_normal_lpdf.hpp.

template<typename ColVec , require_eigen_col_vector_vt< is_fvar, ColVec > * = nullptr>

auto stan::math::softmax ( const ColVec &  alpha )

inline

Definition at line 16 of file softmax.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::softmax ( const var_value< T > &  A )

inline

Returns softmax of given argument.

Template Parameters

T	type of the argument

Parameters

A

argument

Returns

Softmax of the argument

Definition at line 23 of file softmax.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

matrix_cl<double> stan::math::softmax ( const T &  a )

inline

Return the softmax of the specified vector.

Template Parameters

T	type of the vector

Parameters

[in]

a

Vector to transform.

Returns

softmax of the vector

Definition at line 23 of file softmax.hpp.

template<typename Mat , require_rev_matrix_t< Mat > * = nullptr>

auto stan::math::softmax ( const Mat &  alpha )

inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

Parameters

alpha

Unconstrained input vector.

Returns

Softmax of the input.

Exceptions

std::domain_error

If the input vector is size 0.

Definition at line 28 of file softmax.hpp.

template<typename ColVec , require_eigen_col_vector_vt< std::is_arithmetic, ColVec > * = nullptr>

plain_type_t<ColVec> stan::math::softmax ( const ColVec &  v )

inline

Return the softmax of the specified vector.

\( \mbox{softmax}(y) = \frac{\exp(y)} {\sum_{k=1}^K \exp(y_k)}, \)

The entries in the Jacobian of the softmax function are given by \( \begin{array}{l} \displaystyle \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] \\[8pt] \displaystyle \mbox{ } \ \ \ = \left\{ \begin{array}{ll} \mbox{softmax}(y)[k] \times (1 - \mbox{softmax}(y)[m]) & \mbox{ if } m = k, \mbox{ and} \\[6pt] -\mbox{softmax}(y)[k] \times \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. \end{array} \)

Template Parameters

ColVec

type of elements in the vector

Parameters

[in]

v

Vector to transform.

Returns

Unit simplex result of the softmax transform of the vector.

Definition at line 47 of file softmax.hpp.

template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::sort_asc ( T &&  input )

inline

Sort the (row)vector in ascending order.

Parameters

input

vector of row vector to sort

Definition at line 21 of file sort_asc.hpp.

template<typename T >

std::vector<T> stan::math::sort_asc ( std::vector< T >  xs )

inline

Return the specified standard vector in ascending order.

Template Parameters

T	Type of elements contained in vector.

Parameters

xs	Vector to order.

Returns

Vector in ascending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 22 of file sort_asc.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

plain_type_t<EigVec> stan::math::sort_asc ( EigVec &&  xs )

inline

Return the specified vector in ascending order.

Template Parameters

EigVec

type of the vector

Parameters

xs	Vector to order.

Returns

Vector in ascending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 38 of file sort_asc.hpp.

template<typename T , require_all_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::sort_desc ( T &&  input )

inline

Sort the (row)vector in ascending order.

Parameters

input

vector of row vector to sort

Definition at line 21 of file sort_desc.hpp.

template<typename T >

std::vector<T> stan::math::sort_desc ( std::vector< T >  xs )

inline

Return the specified standard vector in descending order.

Template Parameters

T	Type of elements contained in vector.

Parameters

xs	Vector to order.

Returns

Vector in descending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 23 of file sort_desc.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

plain_type_t<EigVec> stan::math::sort_desc ( EigVec &&  xs )

inline

Return the specified vector in descending order.

Template Parameters

EigVec

type of the vector

Parameters

xs	Vector to order.

Returns

Vector in descending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 39 of file sort_desc.hpp.

template<bool ascending, typename C >

std::vector<int> stan::math::sort_indices

(

const C &

xs

)

Return an integer array of indices of the specified container sorting the values in ascending or descending order based on the value of the first template parameter.

Template Parameters

ascending	true if sort is in ascending order
C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 64 of file sort_indices.hpp.

template<typename C >

std::vector<int> stan::math::sort_indices_asc

(

const C &

xs

)

Return a sorted copy of the argument container in ascending order.

Template Parameters

C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 21 of file sort_indices_asc.hpp.

template<typename C >

std::vector<int> stan::math::sort_indices_desc

(

const C &

xs

)

Return a sorted copy of the argument container in ascending order.

Template Parameters

C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 21 of file sort_indices_desc.hpp.

template<typename T >

fvar<T> stan::math::sqrt ( const fvar< T > &  x )

inline

Definition at line 17 of file sqrt.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::sqrt ( const var_value< T > &  A )

inline

Returns the elementwise sqrt() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise sqrt() of the input.

Definition at line 20 of file sqrt.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::sqrt ( const std::complex< fvar< T >> &  z )

inline

Return the square root of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

square root of the argument

Definition at line 30 of file sqrt.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>

auto stan::math::sqrt ( const Container &  x )

inline

Vectorized version of sqrt().

Template Parameters

Container

type of container

Parameters

x

container

Returns

Square root of each value in x.

Definition at line 41 of file sqrt.hpp.

var stan::math::sqrt ( const var &  a )

inline

Return the square root of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}\).

\[ \mbox{sqrt}(x) = \begin{cases} \textrm{NaN} & x < 0 \\ \sqrt{x} & \mbox{if } x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sqrt}(x)}{\partial x} = \begin{cases} \textrm{NaN} & x < 0 \\ \frac{1}{2\sqrt{x}} & x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable whose square root is taken.

Returns

Square root of variable.

Definition at line 44 of file sqrt.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr>

auto stan::math::sqrt ( const Container &  x )

inline

Version of sqrt() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Square root of each value in x.

Definition at line 56 of file sqrt.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::sqrt ( const T &  a )

inline

Return elementwise square root of vector.

Template Parameters

T	a var_value with inner Eigen type

Parameters

a

input

Returns

elementwise square root of vector

Definition at line 58 of file sqrt.hpp.

std::complex<var> stan::math::sqrt ( const std::complex< var > &  z )

inline

Return the square root of the complex argument.

Parameters

[in]

z

argument

Returns

square root of the argument

Definition at line 71 of file sqrt.hpp.

static constexpr double stan::math::sqrt2 ( )

inlinestatic

Returns the square root of two.

Returns

Square root of two.

Definition at line 235 of file constants.hpp.

template<typename T >

fvar<T> stan::math::square ( const fvar< T > &  x )

inline

Definition at line 12 of file square.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::square ( const var_value< T > &  A )

inline

Returns the elementwise square() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise square() of the input.

Definition at line 20 of file square.hpp.

double stan::math::square ( double  x )

inline

Return the square of the specified argument.

\(\mbox{square}(x) = x^2\).

The implementation of square(x) is just x * x. Given this, this method is mainly useful in cases where x is not a simple primitive type, particularly when it is an autodiff type.

Parameters

x	Input to square.

Returns

Square of input.

Definition at line 26 of file square.hpp.

var stan::math::square ( const var &  x )

inline

Return the square of the input variable.

Using square(x) is more efficient than using x * x.

\[ \mbox{square}(x) = \begin{cases} x^2 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{square}(x)}{\partial x} = \begin{cases} 2x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x	Variable to square.

Returns

Square of variable.

Definition at line 35 of file square.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::square ( const T &  x )

inline

Return the elementwise square of x.

Template Parameters

T

type of x

Parameters

x

argument

Returns

elementwise square of x

Definition at line 49 of file square.hpp.

template<typename Container , require_not_stan_scalar_t< Container > * = nullptr, require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::square ( const Container &  x )

inline

Vectorized version of square().

Template Parameters

Container

type of container

Parameters

x

container

Returns

Each value in x squared.

Definition at line 54 of file square.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::square ( const Container &  x )

inline

Version of square() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Each value in x squared.

Definition at line 68 of file square.hpp.

var stan::math::squared_distance ( const var &  a, const var &  b  )

inline

Returns the squared distance.

Definition at line 21 of file squared_distance.hpp.

template<typename Scal1 , typename Scal2 , require_all_stan_scalar_t< Scal1, Scal2 > * = nullptr, require_all_not_var_t< Scal1, Scal2 > * = nullptr>

return_type_t<Scal1, Scal2> stan::math::squared_distance ( const Scal1 &  x1, const Scal2 &  x2  )

inline

Returns the squared distance.

Template Parameters

Scal1	Type of the first scalar.
Scal2	Type of the second scalar.

Parameters

x1	First scalar.
x2	Second scalar.

Returns

Squared distance between scalars

Exceptions

std::domain_error

Any scalar is not finite.

Definition at line 26 of file squared_distance.hpp.

template<typename T_a , typename T_b , require_all_kernel_expressions_and_none_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::squared_distance ( const T_a &  a, const T_b &  b  )

inline

Returns the squared distance.

Template Parameters

T_a	type of the first kernel generator expression
T_b	type of the second kernel generator expression

Parameters

a	first kernel generator expression
b	second kernel generator expression

Returns

squared distance.

Exceptions

std::domain_error

If the matrices are not the same size

Definition at line 30 of file squared_distance.hpp.

template<typename T_a , typename T_b , require_all_nonscalar_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr>

var_value<double> stan::math::squared_distance ( T_a &&  a, T_b &&  b  )

inline

Returns the squared distance.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

the squared distance of the input.

Definition at line 32 of file squared_distance.hpp.

var stan::math::squared_distance ( const var &  a, double  b  )

inline

Returns the squared distance.

Definition at line 35 of file squared_distance.hpp.

var stan::math::squared_distance ( double  a, const var &  b  )

inline

Returns the squared distance.

Definition at line 47 of file squared_distance.hpp.

template<typename EigVec1 , typename EigVec2 , require_all_eigen_vector_t< EigVec1, EigVec2 > * = nullptr, require_all_not_eigen_vt< is_var, EigVec1, EigVec2 > * = nullptr>

return_type_t<EigVec1, EigVec2> stan::math::squared_distance ( const EigVec1 &  v1, const EigVec2 &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

EigVec1	type of the first vector (must be derived from Eigen::MatrixBase and have one compile time dimension equal to 1)
EigVec2	type of the second vector (must be derived from Eigen::MatrixBase and have one compile time dimension equal to 1)

Parameters

v1	First vector.
v2	Second vector.

Returns

Square of distance between vectors.

Exceptions

std::domain_error

If the vectors are not the same size.

Definition at line 50 of file squared_distance.hpp.

template<typename EigVecVar1 , typename EigVecVar2 , require_all_eigen_vector_vt< is_var, EigVecVar1, EigVecVar2 > * = nullptr>

var stan::math::squared_distance ( const EigVecVar1 &  v1, const EigVecVar2 &  v2  )

inline

Definition at line 119 of file squared_distance.hpp.

template<typename EigVecVar , typename EigVecArith , require_eigen_vector_vt< is_var, EigVecVar > * = nullptr, require_eigen_vector_vt< std::is_arithmetic, EigVecArith > * = nullptr>

var stan::math::squared_distance ( const EigVecVar &  v1, const EigVecArith &  v2  )

inline

Definition at line 127 of file squared_distance.hpp.

template<typename EigVecArith , typename EigVecVar , require_eigen_vector_vt< std::is_arithmetic, EigVecArith > * = nullptr, require_eigen_vector_vt< is_var, EigVecVar > * = nullptr>

var stan::math::squared_distance ( const EigVecArith &  v1, const EigVecVar &  v2  )

inline

Definition at line 135 of file squared_distance.hpp.

template<typename T1 , typename T2 , require_all_vector_t< T1, T2 > * = nullptr, require_any_var_vector_t< T1, T2 > * = nullptr>

var stan::math::squared_distance ( const T1 &  A, const T2 &  B  )

inline

Compute the squared distance between the elements in two inputs.

This overload handles arguments where one of T1 or T2 are var_value<T> where T is an Eigen type. The other type can also be a var_value or it can be a matrix type that inherits from EigenBase

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

A	first argument
B	second argument

Returns

sum of squared difference of A and B

Definition at line 157 of file squared_distance.hpp.

void stan::math::stan_print ( std::ostream *  o, const var &  x  )

inline

Definition at line 11 of file stan_print.hpp.

template<typename T , require_not_container_t< T > * = nullptr>

void stan::math::stan_print	(	std::ostream *	o,
		const T &	x
	)

Definition at line 13 of file stan_print.hpp.

template<typename EigVec , require_eigen_vector_t< EigVec > * = nullptr>

void stan::math::stan_print	(	std::ostream *	o,
		const EigVec &	x
	)

Definition at line 18 of file stan_print.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_eigen_vector_t< EigMat > * = nullptr>

void stan::math::stan_print	(	std::ostream *	o,
		const EigMat &	x
	)

Definition at line 33 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const std::vector< T > &	x
	)

Definition at line 54 of file stan_print.hpp.

static void stan::math::start_nested ( )

inlinestatic

Record the current position so that recover_memory_nested() can find it.

It is preferred to use the nested_rev_autodiff class for nested autodiff as it handles recovery of memory automatically.

Definition at line 16 of file start_nested.hpp.

template<bool Condition, typename T1 , typename T2 , std::enable_if_t< Condition > * = nullptr>

T1 stan::math::static_select	(	T1 &&	a,
		T2 &&	b
	)

Returns one of the arguments that can be of different type, depending on the compile time condition.

Can often be used in place of c++17 if constexpr.

Template Parameters

Condition	condition
T1	type of the first argument
T2	type of the second argument

Parameters

a	first argument; returned if the Condition is true
b	second argument; returned if the Condition is false

Returns

One of the arguemtns, depending on the Condition

Definition at line 24 of file static_select.hpp.

template<bool Condition, typename T1 , typename T2 , std::enable_if_t<!Condition > * = nullptr>

T2 stan::math::static_select	(	T1 &&	a,
		T2 &&	b
	)

Definition at line 30 of file static_select.hpp.

template<typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>

return_type_t<T_y> stan::math::std_normal_lccdf ( const T_y &  y )

inline

Definition at line 24 of file std_normal_lccdf.hpp.

template<typename T_y , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y > * = nullptr>

return_type_t<T_y> stan::math::std_normal_lcdf ( const T_y &  y )

inline

Definition at line 27 of file std_normal_lcdf.hpp.

template<typename T_y >

return_type_t<T_y> stan::math::std_normal_lpdf ( const T_y &  y )

inline

Definition at line 68 of file std_normal_lpdf.hpp.

var stan::math::step ( const var &  a )

inline

Return the step, or heaviside, function applied to the specified variable (stan).

See step() for the double-based version.

The derivative of the step function is zero everywhere but at 0, so for convenience, it is taken to be everywhere zero,

\(\mbox{step}(x) = 0\).

Parameters

a	Variable argument.

Returns

The constant variable with value 1.0 if the argument's value is greater than or equal to 0.0, and value 0.0 otherwise.

Definition at line 26 of file step.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T stan::math::step ( const T &  y )

inline

The step, or Heaviside, function.

The function is defined by

step(y) = (y < 0.0) ? 0 : 1.

\[ \mbox{step}(x) = \begin{cases} 0 & \mbox{if } x \le 0 \\ 1 & \mbox{if } x \ge 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	type of value

Parameters

y

value

Returns

zero if the value is less than zero, and one otherwise

Definition at line 31 of file step.hpp.

template<typename T , require_container_t< T > * = nullptr>

auto stan::math::step ( const T &  x )

inline

Return the elementwise application of step() to specified argument container.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise step of members of container.

Definition at line 61 of file step.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_cdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 23 of file student_t_cdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lccdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 24 of file student_t_lccdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lcdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 24 of file student_t_lcdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lpdf ( const T_y &  y, const T_dof &  nu, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 154 of file student_t_lpdf.hpp.

template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>

auto stan::math::sub_col ( T_x &&  x, size_t  i, size_t  j, size_t  nrows  )

inline

Return a 1 x ncols subrow starting at (i-1, j-1).

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x	input kernel generator expression.
i	Starting row + 1.
j	Starting column + 1.
nrows	Number of rows in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 23 of file sub_col.hpp.

template<typename T , require_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::sub_col ( const T &  m, size_t  i, size_t  j, size_t  nrows  )

inline

Return a nrows x 1 subcolumn starting at (i-1, j-1).

Template Parameters

T	type of the matrix

Parameters

m	Matrix.
i	Starting row + 1.
j	Starting column + 1.
nrows	Number of rows in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 23 of file sub_col.hpp.

template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>

auto stan::math::sub_row ( T_x &&  x, size_t  i, size_t  j, size_t  ncols  )

inline

Return a 1 x ncols subrow starting at (i-1, j-1).

Template Parameters

T_x	type of input kernel generator expression x

Parameters

x	input kernel generator expression.
i	Starting row + 1.
j	Starting column + 1.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 23 of file sub_row.hpp.

template<typename T , require_matrix_t< T > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::sub_row ( const T &  m, size_t  i, size_t  j, size_t  ncols  )

inline

Return a 1 x ncols subrow starting at (i-1, j-1).

Template Parameters

T	type of the matrix

Parameters

m	Matrix Input matrix.
i	Starting row + 1.
j	Starting column + 1.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 23 of file sub_row.hpp.

template<typename ScalarA , typename ScalarB , require_all_stan_scalar_t< ScalarA, ScalarB > * = nullptr, require_all_not_var_t< ScalarA, ScalarB > * = nullptr>

return_type_t<ScalarA, ScalarB> stan::math::subtract ( const ScalarA &  a, const ScalarB &  b  )

inline

Return the result of subtracting the second scalar from the first scalar.

Template Parameters

ScalarA	type of the first scalar
ScalarB	type of the second scalar

Parameters

a	first scalar
b	second scalar

Returns

difference between first scalar and second scalar

Definition at line 24 of file subtract.hpp.

template<typename T_a , typename T_b , require_all_prim_or_rev_kernel_expression_t< T_a, T_b > * = nullptr, require_any_var_t< T_a, T_b > * = nullptr, require_any_not_stan_scalar_t< T_a, T_b > * = nullptr>

auto stan::math::subtract ( T_a &&  a, T_b &&  b  )

inline

Subtraction of two reverse mode matrices and/or kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

The subtraction of the the second argument from the first

Definition at line 31 of file subtract.hpp.

template<typename Mat1 , typename Mat2 , require_all_eigen_t< Mat1, Mat2 > * = nullptr, require_all_not_st_var< Mat1, Mat2 > * = nullptr>

auto stan::math::subtract ( const Mat1 &  m1, const Mat2 &  m2  )

inline

Return the result of subtracting the second specified matrix from the first specified matrix.

Template Parameters

Mat1	type of the first matrix or expression
Mat2	type of the second matrix or expression

Parameters

m1	First matrix or expression.
m2	Second matrix or expression.

Returns

Difference between first matrix and second matrix.

Definition at line 43 of file subtract.hpp.

template<typename Scal , typename Mat , require_stan_scalar_t< Scal > * = nullptr, require_eigen_t< Mat > * = nullptr, require_all_not_st_var< Mat, Scal > * = nullptr>

auto stan::math::subtract ( const Scal  c, const Mat &  m  )

inline

Return the result of subtracting the specified matrix from the specified scalar.

Template Parameters

Scal	type of the scalar
Mat	type of the matrix or expression

Parameters

c	Scalar.
m	Matrix or expression.

Returns

The scalar minus the matrix.

Definition at line 61 of file subtract.hpp.

template<typename Mat , typename Scal , require_eigen_t< Mat > * = nullptr, require_stan_scalar_t< Scal > * = nullptr, require_all_not_st_var< Scal, Mat > * = nullptr>

auto stan::math::subtract ( const Mat &  m, const Scal  c  )

inline

Return the result of subtracting the specified scalar from the specified matrix.

Template Parameters

Mat	type of the matrix or expression
Scal	type of the scalar

Parameters

m	Matrix or expression.
c	Scalar.

Returns

The matrix minus the scalar.

Definition at line 78 of file subtract.hpp.

template<typename VarMat1 , typename VarMat2 , require_all_rev_matrix_t< VarMat1, VarMat2 > * = nullptr>

auto stan::math::subtract ( const VarMat1 &  a, const VarMat2 &  b  )

inline

Subtraction operator for matrix variables.

Template Parameters

VarMat1	A matrix of vars or a var with an underlying matrix type.
VarMat2	A matrix of vars or a var with an underlying matrix type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 118 of file operator_subtraction.hpp.

template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::subtract ( const VarMat &  a, const Arith &  b  )

inline

Subtraction operator for a matrix variable and arithmetic.

Template Parameters

VarMat	A matrix of vars or a var with an underlying matrix type.
Arith	A type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 149 of file operator_subtraction.hpp.

template<typename Arith , typename VarMat , require_st_arithmetic< Arith > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::subtract ( const Arith &  a, const VarMat &  b  )

inline

Subtraction operator for an arithmetic type and matrix variable.

Template Parameters

VarMat	A matrix of vars or a var with an underlying matrix type.
Arith	A type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 175 of file operator_subtraction.hpp.

template<typename Var , typename EigMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr>

auto stan::math::subtract ( const Var &  a, const EigMat &  b  )

inline

Subtraction operator for an arithmetic matrix and variable.

Template Parameters

Var	A var_value with an underlying arithmetic type.
EigMat	An Eigen Matrix type with an arithmetic Scalar type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 201 of file operator_subtraction.hpp.

template<typename EigMat , typename Var , require_eigen_vt< std::is_arithmetic, EigMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>

auto stan::math::subtract ( const EigMat &  a, const Var &  b  )

inline

Subtraction operator for a variable and arithmetic matrix.

Template Parameters

EigMat	An Eigen Matrix type with an arithmetic Scalar type.
Var	A var_value with an underlying arithmetic type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 220 of file operator_subtraction.hpp.

template<typename Var , typename VarMat , require_var_vt< std::is_arithmetic, Var > * = nullptr, require_rev_matrix_t< VarMat > * = nullptr>

auto stan::math::subtract ( const Var &  a, const VarMat &  b  )

inline

Subtraction operator for a variable and variable matrix.

Template Parameters

VarMat	An Eigen Matrix type with a variable Scalar type or a var_value with an underlying matrix type.
Var	A var_value with an underlying arithmetic type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 240 of file operator_subtraction.hpp.

template<typename Var , typename VarMat , require_rev_matrix_t< VarMat > * = nullptr, require_var_vt< std::is_arithmetic, Var > * = nullptr>

auto stan::math::subtract ( const VarMat &  a, const Var &  b  )

inline

Subtraction operator for a variable matrix and variable.

Template Parameters

VarMat	An Eigen Matrix type with a variable Scalar type or a var_value with an underlying matrix type.
Var	A var_value with an underlying arithmetic type.

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting two variables.

Definition at line 269 of file operator_subtraction.hpp.

template<typename T1 , typename T2 , require_any_var_vt< std::is_arithmetic, T1, T2 > * = nullptr, require_any_arithmetic_t< T1, T2 > * = nullptr>

auto stan::math::subtract ( const T1 &  a, const T2 &  b  )

inline

Definition at line 288 of file operator_subtraction.hpp.

template<typename T1 , typename T2 , require_all_var_vt< std::is_arithmetic, T1, T2 > * = nullptr>

auto stan::math::subtract ( const T1 &  a, const T2 &  b  )

inline

Definition at line 294 of file operator_subtraction.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr>

T stan::math::sum ( T &&  m )

inline

Returns specified input value.

Template Parameters

T	Type of element.

Parameters

m	Specified value.

Returns

Same value (the sum of one value).

Definition at line 21 of file sum.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::sum ( const var_value< T > &  x )

inline

Returns the sum of the coefficients of the specified matrix on the OpenCL device.

Parameters

x	Specified var_value containing a matrix.

Returns

Sum of coefficients of matrix.

Definition at line 22 of file sum.hpp.

template<typename T >

fvar<T> stan::math::sum ( const std::vector< fvar< T >> &  m )

inline

Return the sum of the entries of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

m

Vector.

Returns

Sum of vector entries.

Definition at line 22 of file sum.hpp.

template<typename Alloc >

var stan::math::sum ( const std::vector< var, Alloc > &  m )

inline

Returns the sum of the entries of the specified vector.

Parameters

m

Vector.

Returns

Sum of vector entries.

Definition at line 26 of file sum.hpp.

template<typename T , require_not_var_t< T > * = nullptr>

T stan::math::sum ( const std::vector< T > &  m )

inline

Return the sum of the values in the specified standard vector.

Template Parameters

T	Type of elements summed.

Parameters

m	Standard vector to sum.

Returns

Sum of elements.

Definition at line 33 of file sum.hpp.

template<typename T , require_eigen_vt< is_fvar, T > * = nullptr>

value_type_t< T > stan::math::sum ( const T &  m )

inline

Return the sum of the entries of the specified matrix.

Returns the sum of the coefficients of the specified Eigen Matrix, Array or expression.

Calculates sum of given kernel generator expression.

Template Parameters

T	type of the matrix

Parameters

m

Matrix.

Returns

Sum of matrix entries.

Template Parameters

T	type of the expression

Parameters

m	expression to sum

Returns

sum of given expression

Template Parameters

T	Type of argument

Parameters

m

argument

Returns

Sum of coefficients of argument.

Definition at line 44 of file sum.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

var stan::math::sum ( const T &  x )

inline

Returns the sum of the coefficients of the specified matrix.

Template Parameters

T	type of the matrix of vector. Can be either a var matrix or matrix of vars.

Parameters

x	Specified var_value containing a matrix or vector.

Returns

Sum of coefficients of matrix.

Definition at line 47 of file sum.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::svd_U

(

const EigMat &

m

)

Given input matrix m, return matrix U where m = UDV^{T}

Template Parameters

EigMat

type of the matrix

Parameters

m	MxN input matrix

Returns

Orthogonal matrix U

Definition at line 19 of file svd_U.hpp.

template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>

auto stan::math::svd_U ( const EigMat &  m )

inline

Given input matrix m, return matrix U where m = UDV^{T}

Adjoint update equation comes from Equation (4) in Differentiable Programming Tensor Networks(H. Liao, J. Liu, et al., arXiv:1903.09650).

Template Parameters

EigMat

type of input matrix

Parameters

m	MxN input matrix

Returns

Orthogonal matrix U

Definition at line 25 of file svd_U.hpp.

template<typename EigMat , require_eigen_matrix_dynamic_t< EigMat > * = nullptr, require_not_st_var< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::svd_V

(

const EigMat &

m

)

Given input matrix m, return matrix V where m = UDV^{T}

Template Parameters

EigMat

type of the matrix

Parameters

m	MxN input matrix

Returns

Orthogonal matrix V

Definition at line 19 of file svd_V.hpp.

template<typename EigMat , require_rev_matrix_t< EigMat > * = nullptr>

auto stan::math::svd_V ( const EigMat &  m )

inline

Given input matrix m, return matrix V where m = UDV^{T}

Adjoint update equation comes from Equation (4) in Differentiable Programming Tensor Networks(H. Liao, J. Liu, et al., arXiv:1903.09650).

Template Parameters

EigMat

type of input matrix

Parameters

m	MxN input matrix

Returns

Orthogonal matrix V

Definition at line 25 of file svd_V.hpp.

template<bool need_eigenvectors = true>

void stan::math::symmetric_eigensolver	(	const matrix_cl< double > &	A,
		matrix_cl< double > &	eigenvalues,
		matrix_cl< double > &	eigenvectors
	)

Definition at line 13 of file symmetric_eigensolver.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::symmetrize_from_lower_tri ( const T &  m )

inline

Return a symmetric matrix using elements from the lower triangular part of the input matrix.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 20 of file symmetrize_from_lower_tri.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::symmetrize_from_lower_tri ( T_x &&  x )

inline

Return a symmetric matrix using elements from the upper triangular part of the input matrix.

Template Parameters

T_x	type of elements in the matrix

Parameters

x

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 21 of file symmetrize_from_lower_tri.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::symmetrize_from_lower_tri ( const var_value< T > &  A )

inline

Return a symmetric matrix using elements from the upper triangular part of the input matrix.

Template Parameters

T_x	type of elements in the matrix

Parameters

A

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 22 of file symmetrize_from_lower_tri.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

Eigen::Matrix<value_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::symmetrize_from_upper_tri ( const T &  m )

inline

Return a symmetric matrix using elements from the upper triangular part of the input matrix.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 20 of file symmetrize_from_upper_tri.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::symmetrize_from_upper_tri ( T_x &&  x )

inline

Return a symmetric matrix using elements from the lower triangular part of the input matrix.

Template Parameters

T_x	type of the matrix

Parameters

x

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 21 of file symmetrize_from_upper_tri.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::symmetrize_from_upper_tri ( const var_value< T > &  A )

inline

Return a symmetric matrix using elements from the upper triangular part of the input matrix.

Template Parameters

T_x	type of elements in the matrix

Parameters

A

Matrix.

Exceptions

std:invalid_argument

if the matrix is not square.

Definition at line 22 of file symmetrize_from_upper_tri.hpp.

void stan::math::system_error ( const char *  function, const char *  name, const int &  y, const char *  msg1, const char *  msg2  )

inline

Throw a system error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing system errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Error code.
[in]	msg1	Message to print before the variable.
[in]	msg2	Message to print after the variable.

Exceptions

std::system_error

Always.

Definition at line 25 of file system_error.hpp.

void stan::math::system_error ( const char *  function, const char *  name, const int &  y, const char *  msg1  )

inline

Throw a system error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing system errors. This will allow us to change the behavior for all functions at once. The message is: * "<function>: <name> <msg1><y>"

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Error code.
[in]	msg1	Message to print before the variable.

Exceptions

std::system_error

Always.

Definition at line 45 of file system_error.hpp.

template<typename T , require_vector_t< T > * = nullptr>

auto stan::math::tail ( const T &  v, size_t  n  )

inline

Return the specified number of elements as a vector or row vector (same as input) from the back of the specified vector or row vector.

Template Parameters

T	type of the vector

Parameters

v	Vector input.
n	Size of return.

Returns

The last n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 23 of file tail.hpp.

template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>

auto stan::math::tail ( T_x &&  x, size_t  n  )

inline

Return the specified number of elements as a vector or row vector (same as input) from the back of the specified vector or row vector.

Template Parameters

T_x	type of input kernel generator expression.

Parameters

x	input kernel generator expression.
n	Size of return.

Returns

The first n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 24 of file tail.hpp.

template<typename T >

std::vector<T> stan::math::tail	(	const std::vector< T > &	sv,
		size_t	n
	)

Return the specified number of elements as a standard vector from the back of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

sv	Standard vector.
n	Size of return.

Returns

The last n elements of sv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 41 of file tail.hpp.

template<typename T >

fvar<T> stan::math::tan ( const fvar< T > &  x )

inline

Definition at line 14 of file tan.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::tan ( const var_value< T > &  A )

inline

Returns the elementwise tan() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise tan() of the input, in radians.

Definition at line 21 of file tan.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::tan ( const std::complex< fvar< T >> &  z )

inline

Return the tangent of the complex argument.

Parameters

[in]

z

argument

Returns

tangent of the argument

Definition at line 27 of file tan.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::tan ( const Container &  x )

inline

Vectorized version of tan().

Template Parameters

Container

type of container

Parameters

x	angles in radians

Returns

Tangent of each value in x.

Definition at line 43 of file tan.hpp.

var stan::math::tan ( const var &  a )

inline

Return the tangent of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \tan x = \sec^2 x\).

\[ \mbox{tan}(x) = \begin{cases} \tan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tan}(x)}{\partial x} = \begin{cases} \sec^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Tangent of variable.

Definition at line 44 of file tan.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::tan ( const Container &  x )

inline

Version of tan() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Tangent of each value in x.

Definition at line 57 of file tan.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::tan ( const VarMat &  a )

inline

Return the tangent of a radian-scaled variable (cmath).

Template Parameters

Varmat

a var_value with inner Eigen type

Parameters

a	Variable for radians of angle.

Returns

Tangent of variable.

Definition at line 59 of file tan.hpp.

std::complex<var> stan::math::tan ( const std::complex< var > &  z )

inline

Return the tangent of the complex argument.

Parameters

[in]

z

argument

Returns

tangent of the argument

Definition at line 73 of file tan.hpp.

template<typename T >

fvar<T> stan::math::tanh ( const fvar< T > &  x )

inline

Definition at line 15 of file tanh.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::tanh ( const var_value< T > &  A )

inline

Returns the elementwise tanh() of a var_value<matrix_cl<double>> in radians.

Parameters

A

argument

Returns

Elementwise tanh() of the input, in radians.

Definition at line 21 of file tanh.hpp.

template<typename T >

std::complex<fvar<T> > stan::math::tanh ( const std::complex< fvar< T >> &  z )

inline

Return the hyperbolic tangent of the complex argument.

Template Parameters

T	autodiff value type

Parameters

[in]

z

argument

Returns

hyperbolic tangent of the argument

Definition at line 29 of file tanh.hpp.

var stan::math::tanh ( const var &  a )

inline

Return the hyperbolic tangent of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x}\).

\[ \mbox{tanh}(x) = \begin{cases} \tanh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tanh}(x)}{\partial x} = \begin{cases} \mbox{sech}^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic tangent of variable.

Definition at line 42 of file tanh.hpp.

template<typename Container , require_not_container_st< std::is_arithmetic, Container > * = nullptr, require_not_var_matrix_t< Container > * = nullptr, require_all_not_nonscalar_prim_or_rev_kernel_expression_t< Container > * = nullptr>

auto stan::math::tanh ( const Container &  x )

inline

Vectorized version of tanh().

Template Parameters

Container

type of container

Parameters

x	angles in radians

Returns

Hyperbolic tangent of each value in x.

Definition at line 43 of file tanh.hpp.

template<typename Container , require_container_st< std::is_arithmetic, Container > * = nullptr>

auto stan::math::tanh ( const Container &  x )

inline

Version of tanh() that accepts std::vectors, Eigen Matrix/Array objects or expressions, and containers of these.

Template Parameters

Container

Type of x

Parameters

x

Container

Returns

Hyperbolic tangent of each value in x.

Definition at line 57 of file tanh.hpp.

template<typename VarMat , require_var_matrix_t< VarMat > * = nullptr>

auto stan::math::tanh ( const VarMat &  a )

inline

Return the hyperbolic tangent of elements of a.

Template Parameters

T

type of a

Parameters

a

argument

Returns

elementwise hyperbolic tangent of a

Definition at line 57 of file tanh.hpp.

std::complex<var> stan::math::tanh ( const std::complex< var > &  z )

inline

Return the hyperbolic tangent of the complex argument.

Parameters

[in]

z

argument

Returns

hyperbolic tangent of the argument

Definition at line 70 of file tanh.hpp.

template<typename EigMat , require_eigen_vt< is_fvar, EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, EigMat::RowsAtCompileTime, EigMat::RowsAtCompileTime> stan::math::tcrossprod ( const EigMat &  m )

inline

Definition at line 16 of file tcrossprod.hpp.

template<typename T_A , typename = require_all_kernel_expressions_and_none_scalar_t<T_A>>

matrix_cl<typename std::decay_t<T_A>::Scalar> stan::math::tcrossprod ( T_A &&  A )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Template Parameters

T	type of elements in A

Parameters

A	input matrix

Returns

A * transpose(A)

Definition at line 21 of file tcrossprod.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::tcrossprod ( const var_value< T > &  M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Template Parameters

T	Type of the matrix

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 22 of file tcrossprod.hpp.

template<typename T , require_eigen_vt< std::is_arithmetic, T > * = nullptr>

Eigen::Matrix<value_type_t<T>, T::RowsAtCompileTime, T::RowsAtCompileTime> stan::math::tcrossprod ( const T &  M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Template Parameters

T	type of the matrix (must be derived from Eigen::MatrixBase)

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 23 of file tcrossprod.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::tcrossprod ( const T &  M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Template Parameters

T	Type of the matrix

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 24 of file tcrossprod.hpp.

double stan::math::tgamma ( double  x )

inline

Return the gamma function applied to the specified argument.

Parameters

x

Argument.

Returns

The gamma function applied to argument.

Definition at line 19 of file tgamma.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::tgamma ( const var_value< T > &  A )

inline

Returns the elementwise tgamma() of a var_value<matrix_cl<double>>.

Parameters

A

argument

Returns

Elementwise tgamma() of the input.

Definition at line 20 of file tgamma.hpp.

template<typename T >

fvar<T> stan::math::tgamma ( const fvar< T > &  x )

inline

Return the result of applying the gamma function to the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

x

Argument.

Returns

Gamma function applied to argument.

Definition at line 21 of file tgamma.hpp.

var stan::math::tgamma ( const var &  a )

inline

Return the Gamma function applied to the specified variable (C99).

The derivative with respect to the argument is

\(\frac{d}{dx} \Gamma(x) = \Gamma(x) \Psi^{(0)}(x)\)

where \(\Psi^{(0)}(x)\) is the digamma function.

\[ \mbox{tgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Gamma(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Gamma(x)=\int_0^{\infty} u^{x - 1} \exp(-u) \, du \]

\[ \frac{\partial \, \Gamma(x)}{\partial x} = \Gamma(x)\Psi(x) \]

Parameters

a	Argument to function.

Returns

The Gamma function applied to the specified argument.

Definition at line 48 of file tgamma.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr, require_not_var_matrix_t< T > * = nullptr>

auto stan::math::tgamma ( const T &  x )

inline

Vectorized version of tgamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Gamma function applied to each value in x.

Exceptions

std::domain_error

if any value is 0 or a negative integer

Definition at line 53 of file tgamma.hpp.

template<typename T , require_var_matrix_t< T > * = nullptr>

auto stan::math::tgamma ( const T &  a )

inline

Return elementwise gamma function.

Template Parameters

T	a var_value with inner Eigen type

Parameters

a

input

Returns

elementwise gamma

Definition at line 62 of file tgamma.hpp.

template<typename T >

void stan::math::throw_domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters

T	Type of variable.

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Variable.
[in]	msg1	Message to print before the variable.
[in]	msg2	Message to print after the variable.

Exceptions

std::domain_error

Always.

Definition at line 26 of file throw_domain_error.hpp.

template<typename T >

void stan::math::throw_domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: * "<function>: <name> <msg1><y>"

Template Parameters

T	Type of variable.

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Variable.
[in]	msg1	Message to print before the variable.

Exceptions

std::domain_error

Always.

Definition at line 49 of file throw_domain_error.hpp.

template<typename T , require_eigen_t< T > * = nullptr>

void stan::math::throw_domain_error_mat ( const char *  function, const char *  name, const T &  y, size_t  i, size_t  j, const char *  msg1, const char *  msg2  )

inline

Throw a domain error with a consistently formatted message for matrices.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index, j+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Row index
j	Column index
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::domain_error

Always.

Definition at line 32 of file throw_domain_error_mat.hpp.

template<typename T >

void stan::math::throw_domain_error_mat ( const char *  function, const char *  name, const T &  y, size_t  i, size_t  j, const char *  msg  )

inline

Throw a domain error with a consistently formatted message for matrices.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index, j+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Row index
j	Column index
msg	Message to print before the variable

Exceptions

std::domain_error

Always

Definition at line 66 of file throw_domain_error_mat.hpp.

template<typename T >

void stan::math::throw_domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::domain_error

Always.

Definition at line 31 of file throw_domain_error_vec.hpp.

template<typename T >

void stan::math::throw_domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg	Message to print before the variable

Exceptions

std::domain_error

Always

Definition at line 57 of file throw_domain_error_vec.hpp.

template<typename T , require_matrix_cl_t< T > * = nullptr>

arena_t<T> stan::math::to_arena

(

const T &

a

)

Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered.

This overload is used for matrix_cl. It is converted to a subclass that schedules its destructor to be called when the memory is recovered.

Template Parameters

T	type of scalar

Parameters

a

argument

Returns

argument

Definition at line 25 of file to_arena.hpp.

template<typename T , require_not_same_t< T, arena_t< T >> * = nullptr, require_not_container_t< T > * = nullptr, require_not_matrix_cl_t< T > * = nullptr>

arena_t<T> stan::math::to_arena ( T &&  a )

inline

Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered.

This overload is used for kernel generator expressions. It also handles any other types that do not have a special overload for them.

Template Parameters

T	type of scalar

Parameters

a

argument

Returns

argument

Definition at line 28 of file to_arena.hpp.

template<typename T , require_same_t< T, arena_t< T >> * = nullptr, require_not_matrix_cl_t< T > * = nullptr, require_not_std_vector_t< T > * = nullptr>

std::remove_reference_t<T> stan::math::to_arena ( T &&  a )

inline

Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered.

For types that already have this property (including scalars and var_values) this is a no-op.

Passing in a lvalue reference to objects not using AD stack, such as a matrix_cl is inefficient as they need to be copied in this case.

Template Parameters

T	type of scalar

Parameters

a

argument

Returns

argument

Definition at line 49 of file to_arena.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_same_t< T, arena_t< T >> * = nullptr>

arena_t<T> stan::math::to_arena ( const T &  a )

inline

Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered.

Converts eigen types to arena_matrix.

Template Parameters

T	type of argument

Parameters

a

argument

Returns

argument copied/evaluated on AD stack

Definition at line 67 of file to_arena.hpp.

template<typename T >

std::vector<T, arena_allocator<T> > stan::math::to_arena ( const std::vector< T, arena_allocator< T >> &  a )

inline

Converts given argument into a type that either has any dynamic allocation on AD stack or schedules its destructor to be called when AD stack memory is recovered.

For std vectors that have data already on AD stack this is a shallow copy.

Template Parameters

T	type of scalar

Parameters

a

argument

Returns

argument

Definition at line 82 of file to_arena.hpp.

template<typename T , require_same_t< T, arena_t< T >> * = nullptr>

arena_t< std::vector< T > > stan::math::to_arena ( const std::vector< T > &  a )

inline

Converts given argument into a type that has any dynamic allocation on AD stack.

Std vectors are copied into another std vector with custom allocator that uses AD stack.

This overload works on vectors with simple scalars that don't need to be converthed themselves.

Template Parameters

T	type of argument

Parameters

a

argument

Returns

argument copied on AD stack

Std vectors are copied into another std vector with custom allocator that uses AD stack.

This overload works on vectors with scalars that also need conversion.

Template Parameters

T	type of argument

Parameters

a

argument

Returns

argument copied on AD stack

Definition at line 109 of file to_arena.hpp.

template<bool Condition, typename T , std::enable_if_t<!Condition > * = nullptr>

T stan::math::to_arena_if ( T &&  a )

inline

If the condition is true, converts given argument into a type that has any dynamic allocation on AD stack.

Otherwise returns the argument

Template Parameters

T	type of argument

Parameters

a

argument

Returns

argument copied/evaluated on AD stack

Definition at line 145 of file to_arena.hpp.

template<bool Condition, typename T , std::enable_if_t< Condition > * = nullptr>

arena_t<T> stan::math::to_arena_if ( const T &  a )

inline

Definition at line 150 of file to_arena.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

std::vector<value_type_t<EigMat> > stan::math::to_array_1d ( const EigMat &  matrix )

inline

Definition at line 15 of file to_array_1d.hpp.

template<typename T >

std::vector<T> stan::math::to_array_1d ( const std::vector< T > &  x )

inline

Definition at line 27 of file to_array_1d.hpp.

template<typename T >

std::vector<typename scalar_type<T>::type> stan::math::to_array_1d ( const std::vector< std::vector< T >> &  x )

inline

Definition at line 33 of file to_array_1d.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

std::vector<std::vector<value_type_t<EigMat> > > stan::math::to_array_2d ( const EigMat &  matrix )

inline

Definition at line 13 of file to_array_2d.hpp.

template<typename T = double, typename S = double>

constexpr std::complex<stan::real_return_t<T, S> > stan::math::to_complex ( const T &  re = 0, const S &  im = 0  )

inline

Return a complex value from a real component and an imaginary component.

Default values for both components is 0.

Template Parameters

T	type of real component
S	type of imaginary component

Parameters

[in]	re	real component (default = 0)
[in]	im	imaginary component (default = 0)

Returns

complex value with specified real and imaginary components

Definition at line 22 of file to_complex.hpp.

template<typename T , require_stan_scalar_t< T > * = nullptr, require_not_fvar_t< T > * = nullptr>

fvar<T> stan::math::to_fvar ( const T &  x )

inline

Definition at line 15 of file to_fvar.hpp.

template<typename T , require_fvar_t< scalar_type_t< T >> * = nullptr>

T&& stan::math::to_fvar ( T &&  x )

inline

Specialization of to_fvar for [containers of] fvars.

Parameters

[in,out]

x

A forward automatic differentation variables.

Returns

The input forward automatic differentiation variables.

Definition at line 26 of file to_fvar.hpp.

template<typename T >

std::vector<fvar<T> > stan::math::to_fvar ( const std::vector< T > &  v )

inline

Definition at line 31 of file to_fvar.hpp.

template<typename T >

std::vector<fvar<T> > stan::math::to_fvar ( const std::vector< T > &  v, const std::vector< T > &  d  )

inline

Definition at line 40 of file to_fvar.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_eigen_vt< is_fvar, T > * = nullptr>

promote_scalar_t<fvar<value_type_t<T> >, T> stan::math::to_fvar ( const T &  m )

inline

Definition at line 51 of file to_fvar.hpp.

template<typename T1 , typename T2 , require_all_eigen_t< T1, T2 > * = nullptr, require_vt_same< T1, T2 > * = nullptr>

promote_scalar_t<fvar<value_type_t<T1> >, T1> stan::math::to_fvar ( const T1 &  val, const T2 &  deriv  )

inline

Definition at line 60 of file to_fvar.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

EigMat stan::math::to_matrix ( EigMat &&  x )

inline

Returns a matrix with dynamic dimensions constructed from an Eigen matrix which is either a row vector, column vector, or matrix.

The runtime dimensions will be the same as the input.

Template Parameters

EigMat

type of the matrix

Parameters

x

matrix

Returns

the matrix representation of the input

Definition at line 24 of file to_matrix.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

var_value<matrix_cl<double> > stan::math::to_matrix ( const var_value< T_x > &  x, int  m, int  n  )

inline

Returns a matrix representation of a vector or matrix in column-major order with the specified number of rows and columns.

Template Parameters

T_x	type of the matrix

Parameters

x	matrix
m	rows
n	columns

Returns

Reshaped input matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 27 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< Eigen::Matrix< T, 1, Eigen::Dynamic >> &  x )

inline

Returns a matrix representation of a standard vector of Eigen row vectors with the same dimensions and indexing order.

Template Parameters

T	type of the elements in the vector

Parameters

x	Eigen vector of vectors of scalar values

Returns

the matrix representation of the input

Definition at line 37 of file to_matrix.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

matrix_cl<return_type_t<T_x> > stan::math::to_matrix ( const T_x &  x, int  m, int  n  )

inline

Returns a matrix representation of a vector or matrix in column-major order with the specified number of rows and columns.

Template Parameters

T_x	type of the matrix

Parameters

x	matrix
m	rows
n	columns

Returns

Reshaped input matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 40 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T, double>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< std::vector< T >> &  x )

inline

Returns a matrix representation of the standard vector of standard vectors with the same dimensions and indexing order.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of vectors of scalar values

Returns

the matrix representation of the input

Definition at line 63 of file to_matrix.hpp.

template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>

auto stan::math::to_matrix ( const T_x &  x, int  m, int  n, bool  col_major  ) -> decltype(to_matrix(x, m, n))

inline

Returns a matrix representation of the vector or matrix in column-major or row major order with the specified number of rows and columns.

Template Parameters

T_x	type of the matrix

Parameters

x	matrix
m	rows
n	columns
col_major	column-major indicator if 1, output matrix is transversed in column-major order if 0, output matrix is transversed in row-major order

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 70 of file to_matrix.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( EigMat &&  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

EigMat

type of the matrix

Parameters

x	matrix
m	rows
n	columns

Returns

Reshaped inputted matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 94 of file to_matrix.hpp.

template<typename T >

auto stan::math::to_matrix ( const std::vector< T > &  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of values
m	rows
n	columns

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 116 of file to_matrix.hpp.

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< int > &  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Parameters

x	vector of values
m	rows
n	columns

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 134 of file to_matrix.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( EigMat &&  x, int  m, int  n, bool  col_major  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

EigMat

type of the matrix

Parameters

x	matrix
m	rows
n	columns
col_major	column-major indicator: if 1, output matrix is transversed in column-major order, if 0, output matrix is transversed in row-major order, otherwise function throws std::invalid_argument

Returns

Reshaped inputted matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 165 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T, double>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< T > &  x, int  m, int  n, bool  col_major  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of values
m	rows
n	columns
col_major	column-major indicator: if 1, output matrix is transversed in column-major order, if 0, output matrix is transversed in row-major order, otherwise function throws std::invalid_argument

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 194 of file to_matrix.hpp.

template<typename T >

ref_type_t<T&&> stan::math::to_ref ( T &&  a )

inline

This evaluates expensive Eigen expressions.

If given expression involves no calculations this is a no-op that should be optimized away.

Template Parameters

T	argument type

Parameters

a

argument

Returns

optionally evaluated argument

Definition at line 17 of file to_ref.hpp.

template<typename T >

ref_type_for_opencl_t<T&&> stan::math::to_ref_for_opencl ( T &&  a )

inline

Converts given Eigen expression into one that can be directly copied to an OpenCL device to create matrix_cl.

If given expression can be directly copied, this is a no-op that should be optimized away.

Template Parameters

T	argument type

Parameters

a

argument

Returns

optionally evaluated argument

Definition at line 19 of file to_ref_for_opencl.hpp.

template<bool Cond, typename T , std::enable_if_t<!Cond > * = nullptr>

T stan::math::to_ref_if ( T &&  a )

inline

No-op that should be optimized away.

Template Parameters

Cond	condition
T	argument type

Parameters

a

argument

Returns

argument

Definition at line 29 of file to_ref.hpp.

template<bool Cond, typename T , std::enable_if_t< Cond > * = nullptr>

ref_type_t<T&&> stan::math::to_ref_if ( T &&  a )

inline

If the condition is true, evaluates expensive Eigen expressions.

If given expression involves no calculations this is a no-op that should be optimized away.

Template Parameters

Cond	condition
T	argument type (Eigen expression)

Parameters

a

argument

Returns

argument converted to Eigen::Ref

Definition at line 43 of file to_ref.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, 1, Eigen::Dynamic> stan::math::to_row_vector ( const EigMat &  matrix )

inline

Definition at line 15 of file to_row_vector.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< T > &  vec )

inline

Definition at line 28 of file to_row_vector.hpp.

Eigen::Matrix<double, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< int > &  vec )

inline

Definition at line 34 of file to_row_vector.hpp.

var stan::math::to_var ( double  x )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

x

A scalar value

Returns

An automatic differentiation variable with the input value.

Definition at line 22 of file to_var.hpp.

var& stan::math::to_var ( var &  x )

inline

Specialization of to_var for non-const var input.

Parameters

[in,out]

x

An automatic differentiation variable.

Returns

The input automatic differentiation variable.

Definition at line 30 of file to_var.hpp.

const var& stan::math::to_var ( const var &  x )

inline

Specialization of to_var for const var input.

Parameters

[in,out]

x

An automatic differentiation variable.

Returns

The input automatic differentiation variable.

Definition at line 38 of file to_var.hpp.

std::vector<var> stan::math::to_var ( const std::vector< double > &  v )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

v

A std::vector<double>

Returns

A std::vector<var> with the values set

Definition at line 48 of file to_var.hpp.

const std::vector<var>& stan::math::to_var ( const std::vector< var > &  v )

inline

Specialization of to_var to for const input vector of var.

Returns a var variable from the input

Parameters

[in]

v

A std::vector<var>

Returns

The input std::vector<var>

Definition at line 64 of file to_var.hpp.

std::vector<var>& stan::math::to_var ( std::vector< var > &  v )

inline

Specialization of to_var to for non-const input vector of var.

Returns a var variable from the input

Parameters

[in]

v

A std::vector<var>

Returns

The input std::vector<var>

Definition at line 74 of file to_var.hpp.

matrix_v stan::math::to_var ( const matrix_d &  m )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

m

A Matrix with scalars

Returns

A Matrix with automatic differentiation variables

Definition at line 84 of file to_var.hpp.

matrix_v& stan::math::to_var ( matrix_v &  m )

inline

Specialization of to_var for non-const matrices of vars.

Parameters

[in,out]

m

A matrix of automatic differentation variables.

Returns

The input matrix of automatic differentiation variables.

Definition at line 95 of file to_var.hpp.

const matrix_v& stan::math::to_var ( const matrix_v &  m )

inline

Specialization of to_var for const matrices of vars.

Parameters

[in,out]

m

A matrix of automatic differentation variables.

Returns

The input matrix of automatic differentiation variables.

Definition at line 103 of file to_var.hpp.

vector_v stan::math::to_var ( const vector_d &  v )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

v

A Vector of scalars

Returns

A Vector of automatic differentiation variables with values of v

Definition at line 114 of file to_var.hpp.

const vector_v& stan::math::to_var ( const vector_v &  v )

inline

Specialization of to_var for const column vector of vars.

Parameters

[in,out]

v

A column vector of automatic differentation variables.

Returns

The input column vector of automatic differentiation variables.

Definition at line 125 of file to_var.hpp.

vector_v& stan::math::to_var ( vector_v &  v )

inline

Specialization of to_var for non-const column vector of vars.

Parameters

[in,out]

v

A column vector of automatic differentation variables.

Returns

The input column vector of automatic differentiation variables.

Definition at line 133 of file to_var.hpp.

row_vector_v stan::math::to_var ( const row_vector_d &  rv )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

rv

A row vector of scalars

Returns

A row vector of automatic differentation variables with values of rv.

Definition at line 144 of file to_var.hpp.

const row_vector_v& stan::math::to_var ( const row_vector_v &  rv )

inline

Specialization of to_var for const row vector of vars.

Parameters

[in,out]

rv

A column vector of automatic differentation variables.

Returns

The input row vector of automatic differentiation variables.

Definition at line 155 of file to_var.hpp.

row_vector_v& stan::math::to_var ( row_vector_v &  rv )

inline

Specialization of to_var for non-const row vector of vars.

Parameters

[in,out]

rv

A column vector of automatic differentation variables.

Returns

The input row vector of automatic differentiation variables.

Definition at line 163 of file to_var.hpp.

template<typename T , require_eigen_vt< is_var, T > * = nullptr>

var_value<Eigen::Matrix<double, T::RowsAtCompileTime, T::ColsAtCompileTime> > stan::math::to_var_value

(

const T &

a

)

Converts an Eigen matrix (or vector or row_vector) or expression of vars into var_value.

Adjoint is propagated back to argument in the reverse pass.

Template Parameters

T	type of the input

Parameters

a	matrix to convert

Definition at line 22 of file to_var_value.hpp.

template<typename T , require_var_t< T > * = nullptr>

T stan::math::to_var_value

(

T &&

a

)

This is a no-op for var_values.

Template Parameters

T	type of the input

Parameters

a	matrix to convert

Definition at line 37 of file to_var_value.hpp.

template<typename T >

auto stan::math::to_var_value

(

const std::vector< T > &

a

)

Convert the elements of the std::vector input to var_value types if possible.

Template Parameters

T	type of elemnts of the input vector

Parameters

a	std::vector of elements to convert

Definition at line 49 of file to_var_value.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

Eigen::Matrix<value_type_t<EigMat>, Eigen::Dynamic, 1> stan::math::to_vector ( const EigMat &  matrix )

inline

Definition at line 15 of file to_vector.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr>

auto stan::math::to_vector ( const var_value< EigMat > &  x )

inline

Reshape a var_value<Matrix> to a var_value<ColumnVector>.

Template Parameters

EigMat

Inner type of the var_value that must inherit from Eigen::EigenBase.

Parameters

x	A var whose inner matrix type is to be reshaped to an Column matrix.

Returns

A view of the original x with inner value and adjoint matrices mapped to a column vector.

Definition at line 20 of file to_vector.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< T > &  vec )

inline

Definition at line 28 of file to_vector.hpp.

Eigen::Matrix<double, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< int > &  vec )

inline

Definition at line 34 of file to_vector.hpp.

template<typename T , require_eigen_t< T > * = nullptr, require_not_st_var< T > * = nullptr>

value_type_t<T> stan::math::trace ( const T &  m )

inline

Returns the trace of the specified matrix.

The trace is defined as the sum of the elements on the diagonal. The matrix is not required to be square. Returns 0 if matrix is empty.

Template Parameters

T	type of the elements in the matrix

Parameters

[in]

m

Specified matrix.

Returns

Trace of the matrix.

Definition at line 22 of file trace.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

value_type_t<T> stan::math::trace

(

const T &

m

)

Calculates trace (sum of diagonal) of given kernel generator expression.

Template Parameters

T	type of the expression

Parameters

m	expression to calculate trace of

Returns

trace of given expression

Definition at line 22 of file trace.hpp.

template<typename T , require_rev_matrix_t< T > * = nullptr>

auto stan::math::trace ( const T &  m )

inline

Returns the trace of the specified matrix.

The trace is defined as the sum of the elements on the diagonal. The matrix is not required to be square. Returns 0 if matrix is empty.

Template Parameters

T	type of the elements in the matrix

Parameters

[in]

m

Specified matrix.

Returns

Trace of the matrix.

Definition at line 23 of file trace.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::trace ( const var_value< T > &  x )

inline

Returns the trace (sum of diagonal) of the specified matrix on the OpenCL device.

Parameters

x	Specified var_value containing a matrix.

Returns

trace of matrix

Definition at line 23 of file trace.hpp.

template<typename Td , typename Ta , typename Tb , require_not_col_vector_t< Td > * = nullptr, require_all_matrix_t< Td, Ta, Tb > * = nullptr, require_any_st_var< Td, Ta, Tb > * = nullptr>

var stan::math::trace_gen_inv_quad_form_ldlt ( const Td &  D, LDLT_factor< Ta > &  A, const Tb &  B  )

inline

Compute the trace of an inverse quadratic form premultiplied by a square matrix.

This computes trace(D B^T A^-1 B) where D is a square matrix and the LDLT_factor of A is provided.

Template Parameters

Td	type of the first matrix
Ta	type of matrix in the LDLT_factor
Tb	type of the second matrix

Parameters

D	a square matrix
A	an LDLT_factor
B	a matrix

Returns

The trace of the inverse quadratic form.

Definition at line 33 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename EigMat1 , typename T2 , typename EigMat3 , require_not_col_vector_t< EigMat1 > * = nullptr, require_all_not_st_var< EigMat1, T2, EigMat3 > * = nullptr>

return_type_t<EigMat1, T2, EigMat3> stan::math::trace_gen_inv_quad_form_ldlt ( const EigMat1 &  D, LDLT_factor< T2 > &  A, const EigMat3 &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(D B^T A^-1 B) where D is a square matrix and the LDLT_factor of A is provided.

Template Parameters

EigMat1	type of the first matrix
T2	type of matrix in the LDLT_factor
EigMat3	type of the third matrix

Parameters

D	multiplier
A	LDLT_factor
B	inner term in quadratic form

Returns

trace(D * B^T * A^-1 * B)

Exceptions

std::domain_error	if D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 36 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename EigVec , typename T , typename EigMat , require_col_vector_t< EigVec > * = nullptr, require_all_not_st_var< EigVec, T, EigMat > * = nullptr>

return_type_t<EigVec, T, EigMat> stan::math::trace_gen_inv_quad_form_ldlt ( const EigVec &  D, LDLT_factor< T > &  A, const EigMat &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(diag(D) B^T A^-1 B) where D is the diagonal of a diagonal matrix (diag(D) is the diagonal matrix itself) and the LDLT_factor of A is provided.

Template Parameters

EigVec	type of the diagonal of first matrix
T	type of matrix in the LDLT_factor
EigMat	type of the B matrix

Parameters

D	diagonal of multiplier
A	LDLT_factor
B	inner term in quadratic form

Returns

trace(diag(D) * B^T * A^-1 * B)

Exceptions

std::domain_error

if A cannot be multiplied by B or B cannot be multiplied by diag(D).

Definition at line 69 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename Td , typename Ta , typename Tb , require_col_vector_t< Td > * = nullptr, require_all_matrix_t< Ta, Tb > * = nullptr, require_any_st_var< Td, Ta, Tb > * = nullptr>

var stan::math::trace_gen_inv_quad_form_ldlt ( const Td &  D, const LDLT_factor< Ta > &  A, const Tb &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(diag(D) B^T A^-1 B) where D is the diagonal of a diagonal matrix (diag(D) is the diagonal matrix itself) and the LDLT_factor of A is provided.

Template Parameters

Td	type of the diagonal of first matrix
Ta	type of matrix in the LDLT_factor
Tb	type of the B matrix

Parameters

D	diagonal of multiplier
A	LDLT_factor
B	inner term in quadratic form

Returns

trace(diag(D) * B^T * A^-1 * B)

Exceptions

std::domain_error

if A cannot be multiplied by B or B cannot be multiplied by diag(D).

Definition at line 190 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename TD , typename TA , typename TB , typename = require_all_eigen_t<TD, TA, TB>, typename = require_all_not_vt_var<TD, TA, TB>, typename = require_any_not_vt_arithmetic<TD, TA, TB>>

auto stan::math::trace_gen_quad_form ( const TD &  D, const TA &  A, const TB &  B  )

inline

Return the trace of D times the quadratic form of B and A.

That is, `trace_gen_quad_form(D, A, B) = trace(D * B' * A * B).`

Template Parameters

TD	type of the first matrix or expression
TA	type of the second matrix or expression
TB	type of the third matrix or expression

Parameters

D	multiplier
A	outside term in quadratic form
B	inner term in quadratic form

Returns

trace(D * B' * A * B)

Exceptions

std::domain_error	if A or D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 36 of file trace_gen_quad_form.hpp.

template<typename EigMatD , typename EigMatA , typename EigMatB , require_all_eigen_vt< std::is_arithmetic, EigMatD, EigMatA, EigMatB > * = nullptr>

double stan::math::trace_gen_quad_form ( const EigMatD &  D, const EigMatA &  A, const EigMatB &  B  )

inline

Return the trace of D times the quadratic form of B and A.

That is, `trace_gen_quad_form(D, A, B) = trace(D * B' * A * B).` This is the overload for arithmetic types to allow Eigen's expression templates to be used for efficiency.

Template Parameters

EigMatD	type of the first matrix or expression
EigMatA	type of the second matrix or expression
EigMatB	type of the third matrix or expression

Parameters

D	multiplier
A	outside term in quadratic form
B	inner term in quadratic form

Returns

trace(D * B' * A * B)

Exceptions

std::domain_error	if A or D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 66 of file trace_gen_quad_form.hpp.

template<typename Td , typename Ta , typename Tb , typename = require_any_var_t<value_type_t<Td>, value_type_t<Ta>, value_type_t<Tb>>, typename = require_all_eigen_t<Td, Ta, Tb>>

var stan::math::trace_gen_quad_form ( const Td &  D, const Ta &  A, const Tb &  B  )

inline

Return the trace of D times the quadratic form of B and A.

That is, `trace_gen_quad_form(D, A, B) = trace(D * B' * A * B).`

This overload requires one of D, A, or B to be a var_value<T> where T inherits from EigenBase

Template Parameters

TD	type of first matrix argument
TA	type of second matrix argument
TB	type of third matrix argument

Parameters

D	multiplier
A	outside term in quadratic form
B	inner term in quadratic form

Returns

trace(D * B' * A * B)

Exceptions

std::domain_error	if A or D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 92 of file trace_gen_quad_form.hpp.

template<typename T , typename EigMat2 , typename = require_all_not_st_var<T, EigMat2>>

return_type_t<T, EigMat2> stan::math::trace_inv_quad_form_ldlt ( LDLT_factor< T > &  A, const EigMat2 &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(B^T A^-1 B) where the LDLT_factor of A is provided.

Template Parameters

T	type of the first matrix and the LDLT_factor
EigMat2	type of the second matrix

Parameters

A	first matrix as LDLT
B	second matrix

Definition at line 28 of file trace_inv_quad_form_ldlt.hpp.

template<typename T1 , typename T2 , require_all_matrix_t< T1, T2 > * = nullptr, require_any_st_var< T1, T2 > * = nullptr>

var stan::math::trace_inv_quad_form_ldlt ( LDLT_factor< T1 > &  A, const T2 &  B  )

inline

Compute the trace of an inverse quadratic form premultiplied by a square matrix.

This computes trace(B^T A^-1 B) where the LDLT_factor of A is provided.

Template Parameters

T1	type of elements in the LDLT_factor
T2	type of the second matrix

Parameters

A	an LDLT_factor
B	a matrix

Returns

The trace of the inverse quadratic form.

Definition at line 32 of file trace_inv_quad_form_ldlt.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_t< EigMat1, EigMat2 > * = nullptr, require_any_vt_fvar< EigMat1, EigMat2 > * = nullptr>

return_type_t< EigMat1, EigMat2 > stan::math::trace_quad_form ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Definition at line 18 of file trace_quad_form.hpp.

template<typename EigMat1 , typename EigMat2 , require_all_eigen_vt< std::is_arithmetic, EigMat1, EigMat2 > * = nullptr>

return_type_t<EigMat1, EigMat2> stan::math::trace_quad_form ( const EigMat1 &  A, const EigMat2 &  B  )

inline

Compute trace(B^T A B).

Template Parameters

EigMat1	type of the first matrix
EigMat2	type of the second matrix

Parameters

A	matrix
B	matrix

Returns

The trace of B^T A B

Exceptions

std::domain_error	if A is not square
std::domain_error	if A cannot be multiplied by B

Definition at line 26 of file trace_quad_form.hpp.

template<typename Mat1 , typename Mat2 , require_all_matrix_t< Mat1, Mat2 > * = nullptr, require_any_var_matrix_t< Mat1, Mat2 > * = nullptr>

var stan::math::trace_quad_form ( const Mat1 &  A, const Mat2 &  B  )

inline

Compute trace(B^T A B).

This overload handles arguments where one of Mat1 or Mat2 are var_value<T> where T is an Eigen type. The other type can also be a var_value or it can be a type that inherits from EigenBase

Template Parameters

Mat1	type of the first matrix
Mat2	type of the second matrix

Parameters

A	matrix
B	matrix

Returns

The trace of B^T A B

Exceptions

std::domain_error	if A is not square
std::domain_error	if A cannot be multiplied by B

Definition at line 118 of file trace_quad_form.hpp.

template<typename T , require_matrix_t< T > * = nullptr>

auto stan::math::transpose ( const T &  m )

inline

Transposes a matrix.

Template Parameters

T	type of the matrix or expression

Parameters

m	matrix or expression

Returns

transposed matrix

Definition at line 17 of file transpose.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

auto stan::math::transpose ( const var_value< T > &  M )

inline

Transposes a matrix.

Parameters

M	input matrix

Returns

transposed matrix

Definition at line 19 of file transpose.hpp.

template<typename T >

fvar<T> stan::math::trigamma ( const fvar< T > &  u )

inline

Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument).

Template Parameters

T	inner type of the fvar

Parameters

u

argument

Returns

trigamma function at argument

Definition at line 21 of file trigamma.hpp.

var stan::math::trigamma ( const var &  u )

inline

Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument).

Parameters

u

argument

Returns

trigamma function at argument

Definition at line 23 of file trigamma.hpp.

double stan::math::trigamma ( double  u )

inline

Return the second derivative of the log Gamma function evaluated at the specified argument.

\[ \mbox{trigamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi_1(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trigamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi_1(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi_1(x)=\sum_{n=0}^\infty \frac{1}{(x+n)^2} \]

\[ \frac{\partial \, \Psi_1(x)}{\partial x} = -2\sum_{n=0}^\infty \frac{1}{(x+n)^3} \]

Parameters

[in]

u

argument

Returns

second derivative of log Gamma function at argument

Definition at line 120 of file trigamma.hpp.

double stan::math::trigamma ( int  u )

inline

Return the second derivative of the log Gamma function evaluated at the specified argument.

Parameters

[in]

u

argument

Returns

second derivative of log Gamma function at argument

Definition at line 129 of file trigamma.hpp.

template<typename T , require_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::trigamma ( const T &  x )

inline

Return the elementwise application of trigamma() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise trigamma of container elements

Definition at line 161 of file trigamma.hpp.

template<typename T >

T stan::math::trigamma_impl ( const T &  x )

inline

Return the trigamma function applied to the argument.

The trigamma function returns the second derivative of the log Gamma function evaluated at the specified argument. This base templated version is used in prim, fwd, and rev implementations.

Template Parameters

T	scalar argument type

Parameters

x

argument

Returns

second derivative of log Gamma function at argument

Definition at line 37 of file trigamma.hpp.

template<typename T >

fvar<T> stan::math::trunc ( const fvar< T > &  x )

inline

Return the nearest integral value that is not larger in magnitude than the specified argument.

Template Parameters

T	inner type of the fvar

Parameters

[in]

x

Argument.

Returns

The truncated argument.

Definition at line 20 of file trunc.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::trunc ( const var_value< T > &  A )

inline

Returns the elementwise trunc() of the input var_value<matrix_cl<double>>.

Parameters

A	input var_value<matrix_cl<double>>.

Returns

Elementwise trunc() of the input argument.

Definition at line 21 of file trunc.hpp.

template<typename T , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T > * = nullptr>

auto stan::math::trunc ( const T &  x )

inline

Return the elementwise application of trunc() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise trunc of container elements

Definition at line 42 of file trunc.hpp.

var stan::math::trunc ( const var &  a )

inline

Returns the truncation of the specified variable (C99).

See trunc() for the double-based version.

The derivative is zero everywhere but at integer values, so for convenience the derivative is defined to be everywhere zero,

\(\frac{d}{dx} \mbox{trunc}(x) = 0\).

\[ \mbox{trunc}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trunc}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Truncation of the variable.

Definition at line 43 of file trunc.hpp.

template<typename T >

std::string stan::math::type_str ( )

inline

Determines a string name of a type.

Unsupported types fail static assert.

Returns

name of the type

Definition at line 14 of file type_str.hpp.

template<>

std::string stan::math::type_str< bool > ( )

inline

Definition at line 27 of file type_str.hpp.

template<>

std::string stan::math::type_str< char > ( )

inline

Definition at line 26 of file type_str.hpp.

template<>

std::string stan::math::type_str< double > ( )

inline

Definition at line 24 of file type_str.hpp.

template<>

std::string stan::math::type_str< int > ( )

inline

Definition at line 25 of file type_str.hpp.

template<typename T , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< U > * = nullptr>

auto stan::math::ub_constrain ( T &&  x, U &&  ub  )

inline

Return the upper-bounded value for the specified unconstrained matrix and upper bound.

The transform is

\(f(x) = U - \exp(x)\)

where \(U\) is the upper bound.

Template Parameters

T	type of Matrix
U	type of upper bound

Parameters

[in]	x	free Matrix.
[in]	ub	upper bound

Returns

matrix constrained to have upper bound

Definition at line 32 of file ub_constrain.hpp.

template<typename T , typename U , require_all_stan_scalar_t< T, U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub  )

inline

Return the upper-bounded value for the specified unconstrained matrix and upper bound.

Specialization of ub_constrain to apply a matrix of upper bounds elementwise to each input element.

The transform is

\(f(x) = U - \exp(x)\)

where \(U\) is the upper bound.

Template Parameters

T	type of Matrix
U	type of upper bound

Parameters

[in]	x	free Matrix.
[in]	ub	upper bound

Returns

matrix constrained to have upper bound

Template Parameters

T	A type inheriting from EigenBase.
U	A type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

upper-bound constrained value corresponding to inputs

The transform is

\(f(x) = U - \exp(x)\)

where \(U\) is the upper bound.

Template Parameters

T	Scalar
U	Scalar

Parameters

[in]	x	free Matrix.
[in]	ub	upper bound

Returns

matrix constrained to have upper bound

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
U	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

upper-bound constrained value corresponding to inputs

Definition at line 36 of file ub_constrain.hpp.

template<typename T_x , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_ub > * = nullptr, require_any_var_t< T_x, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_ub > * = nullptr>

var_value<matrix_cl<double> > stan::math::ub_constrain ( T_x &&  x, T_ub &&  ub  )

inline

Return the upper-bounded value for the specified unconstrained input and specified upper bound.

The transform applied is

\(f(x) = U - \exp(x)\)

where \(U\) is the constant upper bound.

Template Parameters

T_x	kernel generator expression
T_ub	kernel generator expression

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound

Returns

constrained matrix

Definition at line 39 of file ub_constrain.hpp.

template<typename T , typename U , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr, require_all_kernel_expressions_t< U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub, return_type_t< T, U > &  lp  )

inline

Return the upper-bounded value for the specified unconstrained matrix and upper bound.

Specialization of ub_constrain to apply a matrix of upper bounds elementwise to each input element.

Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is

\(f(x) = U - \exp(x)\)

where \(U\) is the upper bound.

Template Parameters

T	type of Matrix
U	type of upper bound

Parameters

[in]	x	free Matrix.
[in]	ub	upper bound
[in,out]	lp	reference to log probability to increment

Returns

matrix constrained to have upper bound

The transform is as specified for ub_constrain(T, double). The log absolute Jacobian determinant is

\( \log | \frac{d}{dx} -\mbox{exp}(x) + U | = \log | -\mbox{exp}(x) + 0 | = x\).

Template Parameters

T	type of scalar
U	type of upper bound

Parameters

[in]	x	free scalar
[in]	ub	upper bound
[in,out]	lp	log density

Returns

scalar constrained to have upper bound

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
U	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

upper-bound constrained value corresponding to inputs

Definition at line 60 of file ub_constrain.hpp.

template<typename T , typename U , require_all_stan_scalar_t< T, U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub, std::decay_t< return_type_t< T, U >> &  lp  )

inline

Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

Specialization of ub_constrain to apply a matrix of upper bounds elementwise to each input element.

The transform is as specified for ub_constrain(T, double). The log absolute Jacobian determinant is

\( \log | \frac{d}{dx} -\mbox{exp}(x) + U | = \log | -\mbox{exp}(x) + 0 | = x\).

Template Parameters

T	type of scalar
U	type of upper bound
S	type of log probability

Parameters

[in]	x	free scalar
[in]	ub	upper bound
[in,out]	lp	log density

Returns

scalar constrained to have upper bound

Template Parameters

T	A type inheriting from EigenBase.
U	A type inheriting from EigenBase.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

upper-bound constrained value corresponding to inputs

Definition at line 67 of file ub_constrain.hpp.

template<typename T_x , typename T_ub , require_all_prim_or_rev_kernel_expression_t< T_x, T_ub > * = nullptr, require_any_var_t< T_x, T_ub > * = nullptr, require_any_not_stan_scalar_t< T_x, T_ub > * = nullptr>

var_value<matrix_cl<double> > stan::math::ub_constrain ( T_x &&  x, T_ub &&  ub, var &  lp  )

inline

Return the upper-bounded value for the specified unconstrained input and specified upper bound.

The transform applied is

\(f(x) = U - \exp(x)\)

where \(U\) is the constant upper bound.

Template Parameters

T_x	kernel generator expression
T_ub	kernel generator expression

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound
[in,out]	lp	reference to log probability to increment

Returns

constrained matrix

Definition at line 75 of file ub_constrain.hpp.

template<typename T , typename U , require_eigen_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub  )

inline

Specialization of ub_constrain to apply a scalar upper bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

upper-bound constrained value corresponding to inputs

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

upper-bound constrained value corresponding to inputs

Definition at line 90 of file ub_constrain.hpp.

template<typename T , typename U , require_eigen_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_all_not_st_var< T, U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub, std::decay_t< return_type_t< T, U >> &  lp  )

inline

Specialization of ub_constrain to apply a scalar upper bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

upper-bound constrained value corresponding to inputs

Definition at line 108 of file ub_constrain.hpp.

template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::ub_constrain ( const std::vector< T > &  x, const U &  ub  )

inline

Specialization of ub_constrain to apply a scalar upper bound elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 163 of file ub_constrain.hpp.

template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::ub_constrain ( const std::vector< T > &  x, const U &  ub, return_type_t< T, U > &  lp  )

inline

Specialization of ub_constrain to apply a scalar upper bound elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 183 of file ub_constrain.hpp.

template<typename T , typename U , require_matrix_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr, require_any_st_var< T, U > * = nullptr>

auto stan::math::ub_constrain ( const T &  x, const U &  ub, return_type_t< T, U > &  lp  )

inline

Specialization of ub_constrain to apply a scalar upper bound elementwise to each input.

Template Parameters

T	A type inheriting from EigenBase or a var_value with inner type inheriting from EigenBase.
U	Scalar.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

upper-bound constrained value corresponding to inputs

Definition at line 187 of file ub_constrain.hpp.

template<typename T , typename U >

auto stan::math::ub_constrain ( const std::vector< T > &  x, const std::vector< U > &  ub  )

inline

Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
U	A type inheriting from EigenBase or a standard vector.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output

Returns

lower-bound constrained value corresponding to inputs

Definition at line 203 of file ub_constrain.hpp.

template<typename T , typename U >

auto stan::math::ub_constrain ( const std::vector< T > &  x, const std::vector< U > &  ub, return_type_t< T, U > &  lp  )

inline

Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element.

Template Parameters

T	A Any type with a Scalar scalar_type.
U	A type inheriting from EigenBase or a standard vector.

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	reference to log probability to increment

Returns

lower-bound constrained value corresponding to inputs

Definition at line 224 of file ub_constrain.hpp.

template<bool Jacobian, typename T , typename U >

auto stan::math::ub_constrain ( const T &  x, const U &  ub, return_type_t< T, U > &  lp  )

inline

Specialization of ub_constrain to apply a container of upper bounds elementwise to each input element.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar
U	A type inheriting from Eigen::EigenBase, a var_value with inner type inheriting from Eigen::EigenBase, a standard vector, or a scalar

Parameters

[in]	x	unconstrained input
[in]	ub	upper bound on output
[in,out]	lp	log density accumulator

Returns

lower-bound constrained value corresponding to inputs

Definition at line 253 of file ub_constrain.hpp.

template<typename T , typename U , require_not_std_vector_t< T > * = nullptr, require_stan_scalar_t< U > * = nullptr>

auto stan::math::ub_free ( T &&  y, U &&  ub  )

inline

Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound.

The transform is the reverse of the ub_constrain(T, double) transform,

\(f^{-1}(y) = \log -(y - U)\)

where \(U\) is the upper bound.

Template Parameters

T	type of scalar
U	type of upper bound

Parameters

y	constrained scalar with specified upper bound
ub	upper bound

Returns

unconstrained scalar with respect to upper bound

Exceptions

std::invalid_argument

if constrained scalar is greater than the upper bound.

Definition at line 35 of file ub_free.hpp.

template<typename T , typename U , require_all_eigen_t< T, U > * = nullptr>

auto stan::math::ub_free ( T &&  y, U &&  ub  )

inline

Return the free matrix that corresponds to the specified upper-bounded matrix with respect to the specified upper bound.

The transform is the reverse of the ub_constrain transform.

Template Parameters

T	type of matrix
U	type of upper bound

Parameters

y	constrained matrix with specified upper bound
ub	upper bound

Returns

unconstrained matrix with respect to upper bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the upper bound.

Definition at line 63 of file ub_free.hpp.

template<typename T , typename U , require_not_std_vector_t< U > * = nullptr>

auto stan::math::ub_free ( const std::vector< T >  y, const U &  ub  )

inline

Return the free variable that corresponds to the specified upper-bounded variable with respect to the specified upper bound.

The transform is the reverse of the ub_constrain transform.

Template Parameters

T	type of constrained variable
U	type of upper bound

Parameters

y	constrained variable with specified upper bound
ub	upper bound

Returns

unconstrained variable with respect to upper bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the upper bound.

Definition at line 91 of file ub_free.hpp.

template<typename T , typename U >

auto stan::math::ub_free ( const std::vector< T >  y, const std::vector< U > &  ub  )

inline

Return the free variable that corresponds to the specified upper-bounded variable with respect to the specified upper bound.

The transform is the reverse of the ub_constrain transform.

Template Parameters

T	type of constrained variable
U	type of upper bound

Parameters

y	constrained variable with specified upper bound
ub	upper bound

Returns

unconstrained variable with respect to upper bound

Exceptions

std::invalid_argument

if any element of constrained variable is greater than the upper bound.

Definition at line 115 of file ub_free.hpp.

template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>

return_type_t<T_y, T_low, T_high> stan::math::uniform_cdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 23 of file uniform_cdf.hpp.

template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>

return_type_t<T_y, T_low, T_high> stan::math::uniform_lccdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 25 of file uniform_lccdf.hpp.

template<typename T_y , typename T_low , typename T_high , require_all_not_nonscalar_prim_or_rev_kernel_expression_t< T_y, T_low, T_high > * = nullptr>

return_type_t<T_y, T_low, T_high> stan::math::uniform_lcdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 25 of file uniform_lcdf.hpp.

template<typename T_y , typename T_low , typename T_high >

return_type_t<T_y, T_low, T_high> stan::math::uniform_lpdf ( const T_y &  y, const T_low &  alpha, const T_high &  beta  )

inline

Definition at line 117 of file uniform_lpdf.hpp.

auto stan::math::uniform_simplex ( int  K )

inline

Return a uniform simplex of size K.

Parameters

K	size of the simplex

Returns

A vector of size K with all elements initialised to 1 / K, so that their sum is equal to 1.

Exceptions

std::domain_error

if K is not positive.

Definition at line 18 of file uniform_simplex.hpp.

template<typename EigMat , require_eigen_col_vector_vt< is_fvar, EigMat > * = nullptr>

auto stan::math::unit_vector_constrain ( const EigMat &  y )

inline

Definition at line 20 of file unit_vector_constrain.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::unit_vector_constrain ( const var_value< T > &  A )

inline

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

T	type of the value of input matrix

Parameters

A	vector of K unrestricted variables

Returns

Unit length vector

Definition at line 22 of file unit_vector_constrain.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::unit_vector_constrain ( T_x &&  x )

inline

Return the unit length vector corresponding to the given free vector.

See the Wikipedia page on generating random points on an N-sphere.

Template Parameters

T_x	type of the vector

Parameters

x	vector of unrestricted variables

Returns

Unit length vector

Definition at line 24 of file unit_vector_constrain.hpp.

template<typename T , require_eigen_col_vector_t< T > * = nullptr, require_not_vt_autodiff< T > * = nullptr>

plain_type_t<T> stan::math::unit_vector_constrain ( const T &  y )

inline

Return the unit length vector corresponding to the free vector y.

See the Wikipedia page on generating random points on an N-sphere.

Template Parameters

T	type inheriting from EigenBase that does not have an fvar scalar type.

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Definition at line 27 of file unit_vector_constrain.hpp.

template<typename T , require_rev_col_vector_t< T > * = nullptr>

auto stan::math::unit_vector_constrain ( const T &  y )

inline

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

EigMat

type inheriting from EigenBase that has a var scalar type.

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Definition at line 30 of file unit_vector_constrain.hpp.

template<typename EigMat , typename T , require_eigen_vt< is_fvar, EigMat > * = nullptr, require_stan_scalar_t< T > * = nullptr>

auto stan::math::unit_vector_constrain ( const EigMat &  y, T &  lp  )

inline

Definition at line 44 of file unit_vector_constrain.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var_value<matrix_cl<double> > stan::math::unit_vector_constrain ( const var_value< T > &  A, var &  lp  )

inline

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

T	type of the value of input matrix

Parameters

A	vector of K unrestricted variables
lp	Log probability reference to increment.

Returns

Unit length vector

Definition at line 45 of file unit_vector_constrain.hpp.

template<typename T_x , require_all_kernel_expressions_and_none_scalar_t< T_x > * = nullptr>

auto stan::math::unit_vector_constrain ( T_x &&  x, double &  lp  )

inline

Return the unit length vector corresponding to the given free vector.

See the Wikipedia page on generating random points on an N-sphere.

Template Parameters

T_x	type of the vector

Parameters

	x	vector of unrestricted variables
[in,out]	lp	log probability to increment

Returns

Unit length vector

Definition at line 46 of file unit_vector_constrain.hpp.

template<typename T1 , typename T2 , require_eigen_col_vector_t< T1 > * = nullptr, require_all_not_vt_autodiff< T1, T2 > * = nullptr>

plain_type_t<T1> stan::math::unit_vector_constrain ( const T1 &  y, T2 &  lp  )

inline

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

T1	type inheriting from EigenBase that does not have an fvar scalar type.

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Parameters

lp	Log probability reference to increment.

Definition at line 49 of file unit_vector_constrain.hpp.

template<typename T , require_eigen_col_vector_vt< is_var, T > * = nullptr>

auto stan::math::unit_vector_constrain ( const T &  y, var &  lp  )

inline

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

EigMat

type inheriting from EigenBase that has a var scalar type.

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Parameters

lp	Log probability reference to increment.

Definition at line 61 of file unit_vector_constrain.hpp.

template<bool Jacobian, typename T , require_not_std_vector_t< T > * = nullptr>

auto stan::math::unit_vector_constrain ( const T &  y, return_type_t< T > &  lp  )

inline

Return the unit length vector corresponding to the free vector y.

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	y	vector of K unrestricted variables
[in,out]	lp	log density accumulator

Returns

Unit length vector of dimension K

If the Jacobian parameter is true, the log density accumulator is incremented with the log absolute Jacobian determinant of the transform. All of the transforms are specified with their Jacobians in the Stan Reference Manual chapter Constraint Transforms.

Template Parameters

Jacobian	if true, increment log density accumulator with log absolute Jacobian determinant of constraining transform
T	A standard vector with inner type inheriting from Eigen::DenseBase or a var_value with inner type inheriting from Eigen::DenseBase with compile time dynamic rows and 1 column

Parameters

	y	vector of K unrestricted variables
[in,out]	lp	log density accumulator

Returns

Unit length vector of dimension K

Definition at line 76 of file unit_vector_constrain.hpp.

template<typename EigVec , require_eigen_col_vector_t< EigVec > * = nullptr>

auto stan::math::unit_vector_free ( EigVec &&  x )

inline

Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1.

Thus, the transformation is just the identity

Template Parameters

EigVec

A type derived from EigenBase with 1 compile time row or column.

Parameters

x	unit vector of dimension K

Returns

Unit vector of dimension K considered "free"

Definition at line 24 of file unit_vector_free.hpp.

template<typename T , require_std_vector_t< T > * = nullptr>

auto stan::math::unit_vector_free

(

const T &

x

)

Overload of unit_vector_free() to untransform each Eigen vector in a standard vector.

Template Parameters

T	A standard vector with with a value_type which inherits from Eigen::MatrixBase with compile time rows or columns equal to 1.

Parameters

x	The standard vector to untransform.

Definition at line 39 of file unit_vector_free.hpp.

std::vector<int> stan::math::unitspaced_array ( int  low, int  high  )

inline

Return an array of integers in an ordered sequence.

This produces an array from low to high (included).

Parameters

low	smallest integer
high	largest integer

Returns

An array of size (high - low + 1) with elements linearly spaced between low and high.

Exceptions

std::domain_error

if high is less than low.

Definition at line 23 of file unitspaced_array.hpp.

void stan::math::validate_non_negative_index ( const char *  var_name, const char *  expr, int  val  )

inline

Definition at line 12 of file validate_non_negative_index.hpp.

void stan::math::validate_positive_index ( const char *  var_name, const char *  expr, int  val  )

inline

Check that simplex is at least size 1.

Parameters

var_name	Name of simplex variable
expr	Expression in which variable is declared
val	Size of simplex

Exceptions

std::invalid_argument

if simplex size is less than 1

Definition at line 20 of file validate_positive_index.hpp.

void stan::math::validate_unit_vector_index ( const char *  var_name, const char *  expr, int  val  )

inline

Check that unit vector is at least size 2.

Parameters

var_name	Name of unit vector variable
expr	Expression in which variable is declared
val	Size of unit vector

Exceptions

std::invalid_argument

if simplex size is less than 2

Definition at line 20 of file validate_unit_vector_index.hpp.

template<typename T >

T stan::math::value_of ( const fvar< T > &  v )

inline

Return the value of the specified variable.

Template Parameters

T	inner type of the fvar

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 18 of file value_of.hpp.

template<typename T , require_st_arithmetic< T > * = nullptr>

T stan::math::value_of ( T &&  x )

inline

Inputs that are arithmetic types or containers of airthmetric types are returned from value_of unchanged.

Template Parameters

T	Input type

Parameters

[in]

x

Input argument

Returns

Forwarded input argument

Definition at line 21 of file value_of.hpp.

template<typename T >

auto& stan::math::value_of ( const var_value< T > &  v )

inline

Return the value of the specified variable.

This function is used internally by autodiff functions along with value_of(T x) to extract the double value of either a scalar or an autodiff variable. This function will be called when the argument is a var even if the function is not referred to by namespace because of argument-dependent lookup.

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 24 of file value_of.hpp.

template<typename T , require_complex_t< T > * = nullptr, require_t< std::is_arithmetic< typename std::decay_t< T >::value_type >> * = nullptr>

auto stan::math::value_of ( T &&  x )

inline

Definition at line 28 of file value_of.hpp.

template<typename T , require_std_vector_t< T > * = nullptr, require_not_st_arithmetic< T > * = nullptr>

auto stan::math::value_of ( const T &  x )

inline

For std::vectors of non-arithmetic types, return a std::vector composed of value_of applied to each element.

Template Parameters

T	Input element type

Parameters

[in]

x

Input std::vector

Returns

std::vector of values

Definition at line 50 of file value_of.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_st_arithmetic< EigMat > * = nullptr>

auto stan::math::value_of ( EigMat &&  M )

inline

For Eigen matrices and expressions of non-arithmetic types, return an expression that represents the Eigen::Matrix resulting from applying value_of elementwise.

Template Parameters

EigMat

type of the matrix

Parameters

[in]

M

Matrix to be converted

Returns

Matrix of values

Definition at line 72 of file value_of.hpp.

template<typename T >

auto stan::math::value_of_rec ( const var_value< T > &  v )

inline

Return the value of the specified variable.

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 17 of file value_of_rec.hpp.

template<typename T >

double stan::math::value_of_rec ( const fvar< T > &  v )

inline

Return the value of the specified variable.

Template Parameters

T	inner type of the fvar, must implement value_of_rec

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 20 of file value_of_rec.hpp.

template<typename T , typename = require_stan_scalar_t<T>>

double stan::math::value_of_rec ( const T  x )

inline

Return the value of the specified scalar argument converted to a double value.

See the primitive_value function to extract values without casting to double.

This function is meant to cover the primitive types. For types requiring pass-by-reference, this template function should be specialized.

Template Parameters

T	Type of scalar.

Parameters

x	Scalar to convert to double.

Returns

Value of scalar cast to a double.

Definition at line 29 of file value_of_rec.hpp.

double stan::math::value_of_rec ( double  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified value.

Returns

Specified value.

Definition at line 44 of file value_of_rec.hpp.

template<typename T >

std::complex<double> stan::math::value_of_rec ( const std::complex< T > &  x )

inline

Recursively apply value-of to the parts of the argument.

Template Parameters

T	value type of argument

Parameters

[in]

x

argument

Returns

real complex value of argument

Definition at line 54 of file value_of_rec.hpp.

template<typename T , require_not_same_t< double, T > * = nullptr>

std::vector<double> stan::math::value_of_rec ( const std::vector< T > &  x )

inline

Convert a std::vector of type T to a std::vector of doubles.

T must implement value_of_rec. See test/math/fwd/fun/value_of_rec.cpp for fvar and var usage.

Template Parameters

T	Scalar type in std::vector

Parameters

[in]

x

std::vector to be converted

Returns

std::vector of values

Definition at line 69 of file value_of_rec.hpp.

template<typename T , require_std_vector_t< T > * = nullptr, require_vt_same< double, T > * = nullptr>

T stan::math::value_of_rec ( T &&  x )

inline

Return the specified argument.

See value_of_rec(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified std::vector.

Returns

Specified std::vector.

See value_of_rec(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Template Parameters

T	Type of matrix.

Parameters

x	Specified matrix.

Returns

Specified matrix.

Definition at line 91 of file value_of_rec.hpp.

template<typename T , typename = require_not_st_same<T, double>, typename = require_eigen_t<T>>

auto stan::math::value_of_rec ( T &&  M )

inline

Convert a matrix of type T to a matrix of doubles.

T must implement value_of_rec. See test/unit/math/fwd/fun/value_of_test.cpp for fvar and var usage.

Template Parameters

T	Type of matrix

Parameters

[in]

M

Matrix to be converted

Returns

Matrix of values

Definition at line 107 of file value_of_rec.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

var stan::math::variance ( const var_value< T > &  A )

inline

Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero.

Template Parameters

Mat	input matrix type

Parameters

A

input

Returns

sample variance of the input

Definition at line 24 of file variance.hpp.

template<typename EigMat , require_eigen_t< EigMat > * = nullptr, require_not_vt_var< EigMat > * = nullptr>

value_type_t<EigMat> stan::math::variance ( const EigMat &  m )

inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified matrix.

Template Parameters

EigMat

type inheriting from EigenBase that does not have an var scalar type.

Parameters

m

matrix

Returns

sample variance of coefficients

Exceptions

<code>std::invalid_argument</code>

if the matrix has size zero

Definition at line 25 of file variance.hpp.

template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>

double stan::math::variance ( const T &  a )

inline

Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero.

Template Parameters

T	input matrix type

Parameters

a

input

Returns

sample variance of the input

Definition at line 25 of file variance.hpp.

var stan::math::variance ( const std::vector< var > &  v )

inline

Return the sample variance of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters

[in]

v

a vector

Returns

sample variance of specified vector

Definition at line 42 of file variance.hpp.

template<typename StdVec , require_std_vector_t< StdVec > * = nullptr, require_not_vt_var< StdVec > * = nullptr>

value_type_t<StdVec> stan::math::variance ( const StdVec &  v )

inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector.

Template Parameters

StdVec

A standard library vector that does not contain a var.

Parameters

v	specified vector

Returns

sample variance of vector

Exceptions

<code>std::invalid_argument</code>

if the vector has size zero

Definition at line 46 of file variance.hpp.

template<typename EigMat , require_eigen_vt< is_var, EigMat > * = nullptr>

var stan::math::variance

(

const EigMat &

m

)

Return the sample variance of the specified vector, row vector, or matrix.

Raise domain error if size is not greater than zero.

Template Parameters

EigMat

type inheriting from EigenBase that has a var scalar type.

Parameters

[in]

m

input matrix

Returns

sample variance of specified matrix

Definition at line 61 of file variance.hpp.

template<typename Mat , require_var_matrix_t< Mat > * = nullptr>

var stan::math::variance ( const Mat &  x )

inline

Return the sample variance of the var_value matrix Raise domain error if size is not greater than zero.

Template Parameters

Mat	input matrix type

Parameters

[in]

x

a input

Returns

sample variance of input

Definition at line 79 of file variance.hpp.

template<typename Vec , typename... Args>

auto stan::math::vec_concat ( const Vec &  v1, const Args &...  args  )

inline

Get the event stack from a vector of events and other arguments.

Template Parameters

Vec	type of elements in the array
Args	types for variadic arguments

Parameters

v1	event stack to roll up
args	variadic arguments passed down to the next recursion

Returns

Vector of OpenCL events

Definition at line 55 of file vec_concat.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::von_mises_lpdf	(	T_y const &	y,
		T_loc const &	mu,
		T_scale const &	kappa
	)

Definition at line 26 of file von_mises_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::von_mises_lpdf ( T_y const &  y, T_loc const &  mu, T_scale const &  kappa  )

inline

Definition at line 92 of file von_mises_lpdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::weibull_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  sigma  )

inline

Definition at line 117 of file weibull_lpdf.hpp.

template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >

return_type_t<T_y, T_alpha, T_tau, T_beta, T_delta> stan::math::wiener_lpdf ( const T_y &  y, const T_alpha &  alpha, const T_tau &  tau, const T_beta &  beta, const T_delta &  delta  )

inline

Definition at line 210 of file wiener_lpdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::wishart_lpdf ( const T_y &  W, const T_dof &  nu, const T_scale &  S  )

inline

Definition at line 102 of file wishart_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::wishart_rng ( double  nu, const Eigen::MatrixXd &  S, RNG &  rng  )

inline

Definition at line 15 of file wishart_rng.hpp.

void stan::math::zero_adjoints ( )

inlinenoexcept

End of recursion for set_zero_adjoints.

Definition at line 14 of file zero_adjoints.hpp.

template<typename T , require_st_arithmetic< T > * = nullptr>

void stan::math::zero_adjoints ( T &  x )

inlinenoexcept

Do nothing for non-autodiff arguments.

Recursively call zero_adjoints on the rest of the arguments.

Template Parameters

T	type of current argument
Pargs	type of rest of arguments

Parameters

x	current argument
args	rest of arguments to zero

Definition at line 27 of file zero_adjoints.hpp.

void stan::math::zero_adjoints ( var &  x )

inline

Zero the adjoint of the vari in the first argument.

Recursively call zero_adjoints on the rest of the arguments.

Template Parameters

T	type of current argument
Pargs	type of rest of arguments

Parameters

x	current argument
args	rest of arguments to zero

Definition at line 39 of file zero_adjoints.hpp.

template<typename EigMat , require_eigen_vt< is_autodiff, EigMat > * = nullptr>

void stan::math::zero_adjoints ( EigMat &  x )

inline

Zero the adjoints of the varis of every var in an Eigen::Matrix container.

Recursively call zero_adjoints on the rest of the arguments.

Template Parameters

T	type of current argument
Pargs	type of rest of arguments

Parameters

x	current argument
args	rest of arguments to zero

Definition at line 52 of file zero_adjoints.hpp.

template<typename StdVec , require_std_vector_st< is_autodiff, StdVec > * = nullptr>

void stan::math::zero_adjoints ( StdVec &  x )

inline

Zero the adjoints of every element in a vector.

Recursively call zero_adjoints on the rest of the arguments.

Template Parameters

T	type of current argument
Pargs	type of rest of arguments

Parameters

x	current argument
args	rest of arguments to zero

Definition at line 69 of file zero_adjoints.hpp.

std::vector<double> stan::math::zeros_array ( int  K )

inline

Return an array of zeros.

Parameters

K	size of the array

Returns

an array of size K with all elements initialised to 0.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file zeros_array.hpp.

std::vector<int> stan::math::zeros_int_array ( int  K )

inline

Return an integer array of zeros.

Parameters

K	size of the array

Returns

an integer array of size K with all elements initialised to 0.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file zeros_int_array.hpp.

auto stan::math::zeros_row_vector ( int  K )

inline

Return a row vector of zeros.

Parameters

K	size of the row vector

Returns

A row vector of size K with all elements initialised to 0.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file zeros_row_vector.hpp.

auto stan::math::zeros_vector ( int  K )

inline

Return a vector of zeros.

Parameters

K	size of the vector

Returns

A vector of size K with all elements initialised to 0.

Exceptions

std::domain_error

if K is negative.

Definition at line 17 of file zeros_vector.hpp.

Variable Documentation

const double stan::math::CONSTRAINT_TOLERANCE = 1E-8

The tolerance for checking arithmetic bounds in rank and in simplexes.

The default value is 1E-8.

Definition at line 12 of file constraint_tolerance.hpp.

constexpr double stan::math::EPSILON = std::numeric_limits<double>::epsilon()

static

Smallest positive value.

Definition at line 41 of file constants.hpp.

constexpr double stan::math::HALF_LOG_TWO = 0.5 * LOG_TWO

static

The value of half the natural logarithm 2, \( \log(2) / 2 \).

Definition at line 175 of file constants.hpp.

const double stan::math::HALF_LOG_TWO_PI = 0.5 * LOG_TWO_PI

The value of half the natural logarithm \( 2\pi \), \( \log(2\pi) / 2 \).

Definition at line 181 of file constants.hpp.

constexpr double stan::math::INFTY = std::numeric_limits<double>::infinity()

static

Positive infinity.

Definition at line 46 of file constants.hpp.

constexpr double stan::math::INV_SQRT_PI = boost::math::constants::one_div_root_pi<double>()

static

The value of 1 over the square root of \( \pi \), \( 1 / \sqrt{\pi} \).

Definition at line 155 of file constants.hpp.

constexpr double stan::math::INV_SQRT_TWO = boost::math::constants::one_div_root_two<double>()

static

The value of 1 over the square root of 2, \( 1 / \sqrt{2} \).

Definition at line 148 of file constants.hpp.

constexpr double stan::math::INV_SQRT_TWO_PI = boost::math::constants::one_div_root_two_pi<double>()

static

The value of 1 over the square root of \( 2\pi \), \( 1 / \sqrt{2\pi} \).

Definition at line 162 of file constants.hpp.

constexpr double stan::math::lgamma_stirling_diff_useful = 10

Definition at line 18 of file lgamma_stirling_diff.hpp.

const double stan::math::LOG_EPSILON = std::log(EPSILON)

The natural logarithm of machine precision \( \epsilon \), \( \log \epsilon \).

Definition at line 74 of file constants.hpp.

constexpr double stan::math::LOG_HALF = -LOG_TWO

static

The natural logarithm of 0.5, \( \log 0.5 = \log 1 - \log 2 \).

Definition at line 92 of file constants.hpp.

const double stan::math::LOG_PI = std::log(pi())

The natural logarithm of \( \pi \), \( \log \pi \).

Definition at line 80 of file constants.hpp.

const double stan::math::LOG_PI_OVER_FOUR = 0.25 * LOG_PI

The value of one quarter the natural logarithm of \( \pi \), \( \log(\pi) / 4 \).

Definition at line 104 of file constants.hpp.

const double stan::math::LOG_SQRT_PI = std::log(boost::math::constants::root_pi<double>())

The natural logarithm of the square root of \( \pi \), \( \log(sqrt{\pi}) \).

Definition at line 110 of file constants.hpp.

constexpr double stan::math::LOG_TEN = boost::math::constants::ln_ten<double>()

static

The natural logarithm of 10, \( \log 10 \).

Definition at line 116 of file constants.hpp.

constexpr double stan::math::LOG_TWO = boost::math::constants::ln_two<double>()

static

The natural logarithm of 2, \( \log 2 \).

Definition at line 86 of file constants.hpp.

const double stan::math::LOG_TWO_PI = LOG_TWO + LOG_PI

The natural logarithm of 2 plus the natural logarithm of \( \pi \), \( \log(2\pi) \).

Definition at line 98 of file constants.hpp.

constexpr double stan::math::LOG_ZERO = -INFTY

static

The natural logarithm of 0, \( \log 0 \).

Definition at line 68 of file constants.hpp.

const std::string stan::math::MAJOR_VERSION = "4"

Major version number for Stan math library.

Definition at line 22 of file version.hpp.

const std::string stan::math::MINOR_VERSION = "2"

Minor version number for Stan math library.

Definition at line 25 of file version.hpp.

const double stan::math::NEG_LOG_SQRT_TWO_PI = -std::log(SQRT_TWO_PI)

The value of minus the natural logarithm of the square root of \( 2\pi \), \( -\log(\sqrt{2\pi}) \).

Definition at line 187 of file constants.hpp.

constexpr double stan::math::NEGATIVE_INFTY = -INFTY

static

Negative infinity.

Definition at line 51 of file constants.hpp.

constexpr double stan::math::NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()

static

(Quiet) not-a-number value.

Definition at line 56 of file constants.hpp.

opencl_context stan::math::opencl_context

static

Definition at line 435 of file opencl_context.hpp.

const std::string stan::math::PATCH_VERSION = "1"

Patch version for Stan math library.

Definition at line 28 of file version.hpp.

const double stan::math::POISSON_MAX_RATE = std::pow(2.0, 30)

Largest rate parameter allowed in Poisson RNG.

Definition at line 192 of file constants.hpp.

constexpr double stan::math::SQRT_PI = boost::math::constants::root_pi<double>()

static

The value of the square root of \( \pi \), \( \sqrt{\pi} \).

Definition at line 128 of file constants.hpp.

constexpr double stan::math::SQRT_TWO = boost::math::constants::root_two<double>()

static

The value of the square root of 2, \( \sqrt{2} \).

Definition at line 122 of file constants.hpp.

constexpr double stan::math::SQRT_TWO_OVER_SQRT_PI = SQRT_TWO / SQRT_PI

static

The square root of 2 divided by the square root of \( \pi \), \( \sqrt{2} / \sqrt{\pi} \).

Definition at line 141 of file constants.hpp.

constexpr double stan::math::SQRT_TWO_PI = boost::math::constants::root_two_pi<double>()

static

The value of the square root of \( 2\pi \), \( \sqrt{2\pi} \).

Definition at line 135 of file constants.hpp.

constexpr double stan::math::TWO_OVER_SQRT_PI = boost::math::constants::two_div_root_pi<double>()

static

The value of 2 over the square root of \( \pi \), \( 2 / \sqrt{\pi} \).

Definition at line 169 of file constants.hpp.

constexpr double stan::math::TWO_PI = boost::math::constants::two_pi<double>()

static

Twice the value of \( \pi \), \( 2\pi \).

Definition at line 62 of file constants.hpp.